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REESE   LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 
Deceived        MAR  ...1.5  1893        ,  /^    . 
'Otr,      Class  No. 


PRACTICAL  TREATISE 


RAILWAY    CURVES 


LOCATION, 

FOR  YOUNG  ENGINEERS 


^OBTAINING  A  FULL  DESCRIPTION  OP  THE  INSTRUMENTS,  THE  MANNER  OP  ADJUSTING  THEM,  AN* 

tHE   METHODS   OF    PROCEEDING   IN   THE  FIELD, — NEW  AND  SIMPLE  FORMULA  FOR  <»M- 

POUND   AND   REVER8B  CURVING,— RULES   FOR   CALCULATING  EXCAVATION   A>T> 

EMBANKMENT, — STAKING    OUT  WORK,  &C.,  TOGETHER  WITH   TABLES  OF 

NATURAL  SINES  AND  TANGENTS,  RADII,  CHORDS,  ORDINATES, 

AMD  OTHERS  OF  GENERAL  USE  IN  THE  PROFESSION. 


BY 

WILLIAM  F.  SHUNK, 
UNIVERSITY 


CIVIL  ENGINEER. 


PHILADELPHIA: 
HENRY  CAREY  BAIRD  &  CO., 

INDUSTRIAL  PUBLISHERS,  BOOKSELLERS  AND  IMPORTERS, 
810   WALNUT  STREET. 


Entered,  according  to  the  Act  of  Congress,  in  the  year  1854,  by 
E.  H.  BUfLER  ft  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  in  and  for  the  Extern 
District  of  Pennsylvania. 


PREFACE. 


THE  located  line  for  railway  is  a  series  of  curves  and 
straight  lines,  or  tangents.  These  are  first  plotted  to  a 
large  scale  from  data  gathered  on  preliminary  survey.  It 
is  therefore  desirable  that  all  explorations  should  be  made 
with  extreme  care,  as  upon  their  correctness  depend,  in  no 
small  degree,  the  labour  and  time  required  in  location. 
It  were  better  for  accuracy  that  all  angles  should  be  made 
and  recorded  from  the  plates,  and  the  needle  used  only  as 
a  test,  or  check.  Good  chaining  is  indispensable.  Great 
attention,  too,  should  be  given  to  the  proper  use  of  the 
elope  instrument.  By  these  means  a  working  map  can  be 
constructed  in  the  office  upon  which  the  proposed  location, 
grade  lines,  &c.,  may  be  traced  with  tolerable  resemblance 
to  fact.  Still  many  errors  attach  to  both  data  and  map, 
and  these,  together  with  the  unexpected  obstacles  encoun- 
tered in  the  field,  require  ready  knowledge  of  the  means 
for  overcoming  them. 

It  has  been  my  design  to  present  this  knowledge  to  my 
younger  fellows  in  the  profession.  I  have  endeavoured  to 

(3) 


IV  PREFACE. 

do  it  lucidly  and  concisely — without  supposing  unusual  cases 
— without  prolix  proof  or  complex  figuring.  The  problems 
given  are  of  frequent  occurrence,  and  the  tables  appended 
will  be  found  useful  and  correct. 

To  STRICKLAND  KNEASS,  a  gentleman  whose  professional 
abilities  are  well  known,  I  return  thanks  for  valuable 
assistance.  I  would  likewise  make  my  acknowledgments, 
for  useful  suggestions,  to  CHARLES  DELISLE,  an  engineer 
of  high  mathematical  attainment. 

I  am  aware  that  much  more  might  have  been  said — 
much  more  suggested — on  the  subject  of  location ;  but  a 
field  book  being  the  object,  the  compact  plan  precluded 
any  extensive  essay. 

If  the  work  with  brevity  combines  clearness,  and  is 
comprehensive  withal,  it  is  the  work  intended. 

W.  F.  SHUNK. 


CONTENTS. 


1'A.GK 

PREFACE       ........  3 

Explanations      ........  7 

ARTICLE  I. — Of  the  Instruments.     The  Level        .            .            .  It 

Its  adjustment     ......  13 

The  Rod          ..                         ....  14 

Levelling              .            .            .                         .            .  15 

The  Transit    .......  17 

The  Vernier         .            .            .            .            .            .  18 

Adjustment  of  Transit  .  .  .  .  .19 

II. — Preliminary  propositions               ....  20 

III. — To  avoid  an  obstacle  in  tangent        .  .  .  .21 

IV.— Triangulation        ...  22 

V. — Of  calculating  tangents  to  any  degree  of  curvature              .  24 

VI. — To  trace  a  curve  with  transit  and  chain  ...  25 

VII.- — To  triangulate  on  a  curve      ....  2? 

VIII. — To  change  the  origin  of  any  curve,  so  that  it  shall  termi- 
nate in  a  tangent,  parallel  to  a  given  tangent       .  .31 
IX.— To  change  a  P.  C.  C.  with  similar  object.    First.  When  ihe 

second  curve  has  the  smaller  radius           .             .             .  52 

X. — Second.  When  the  last  curve  has  the  larger  radius        .  34 

Synopsis  of  formulae  for  compound  curving          .             .  35 
XL — Having  located  a  compound  curve  terminating  in  any  tan- 
gent, to  find  the  P.  C.  C.  at  which  to  commence  another 
curve  of  given  radius  which  shall  terminate  in  the  same 
tangent.    First.  When  the  latter  curves  have  the  smaller 

radii      .......  3C 

XII. — Second.  When  the  latter  curves  have  the  larger  radii          .  38 

1*  (5) 


VI  CONTENTS. 

PA.GJ* 
ART.   XIII. — To  change  a  P.  R.  C.  so  that  the  second  curve  shall  .ermi- 

nate  in  a  tangent  parallel  to  a  given  tangent        .            .  39 
XIV. — How  to  proceed  when  the  P.  C.  is  inaccessible                .  41 
XV. — To  avoid  obstacles  in  the  line  of  curve         .                         .  43 
XVI. — To  calculate  reverse  curves          .                                     .  45 
XVII. — Having  givon  a  located  curve,  terminating  in  any  given 
tangent,  to  find  where  a  curve  of  different  radius  will  ter- 
minate in  a  parallel  tangent           .                        .            .  46 
XVIII. — Having  a  curve  located  and  terminating  in  a  given  tangent, 
to  find  the  P.  C.  C.  whereat  to  begin  another  curve  of 
given  radius  which  shall  terminate  in  a  parallel  tangent  48 
XIX.— To  locate  a  Y                   .           .           .           .            .  49 
XX. — To  run  a  tangent  to  two  curves         .            .            .            .61 

XXL— Ofordinates 52 

XXII. — To  find  the  radius  corresponding  to  any  chord  and  deflexion 

angle.     Deflexion  and  tangential  distances    .            .  54 
XXIII. — Of  excavation  and  embankment        .            .            .            .56 

XXIV.— Side  staking         ...  60 

TABLES. 

Natural  Sines  and  Tangents      ......  f>3 

Radii             .....                        •            •            .  89 

Long  Chords      ...                        •                        •            .  90 

Ordinates      .....•••  91 

Squares  and  Square  Roots         .•••••  94 

Slopes  and  Distances  for  Topography         .           .           «            ,           »  108 


EXPLANATIONS 


ALL  railway  curves  are  parts  of  circles.  They  are 
designated  generally  from  their  character  as  simple,  com- 
pound, or  reverse  ;  and  specifically  from  the  central  angle 
subtended  by  a  chord  of  100  feet  at  the  circumference, 
this  being  the  length  of  the  chain  in  common  use.  It  is 
found  that  the  circle  described  with  radius  of  5730  feet 
has  a  circumference  of  36,000  feet.  Since  there  are  360° 
in  the  circle,  the  central  angle  subtended  by  a  chord  of 
100  feet  is,  in  this  case,  equal  to  1°,  and  the  curve  is 
named  a  one  degree  curve.  So  likewise  in  a  circle  with 
radius  of  2865  feet,  half  of  5730,  the  central  angle  cor- 
responding to  the  chord  100  is  2°;  the  curve  is  then  called 
a  two  degree  curve. 

The  beginning  of  a  curve  is  called  the  point  of  curva- 
ture, or  simply  the  P.  C.,  and  its  termination  the  point  of 
tangent,  marked  P.  T. 

A  compound  curve  is  composed  of  two  curves  of  different 
radii,  turning  in  the  same  direction,  and  having  a  common 
tangent  at  their  point  of  meeting.  This  point  is  called  the 
point  of  compound  curvature,  or  P.  C.  C. 

(7) 


RAILWAY    CURVES   AND    LOCATION. 

A  reverse  curve  is  composed  of  two  curves  turning  in 
different  directions,  and  having  a  common  tangent  at  their 
point  of  meeting,  which  latter  is  named  the  point  of  re- 
versed curvature,  or  the  P.  II.  C. 

All  sines  and  tangents  made  use  of  in  this  work  are 
from  the  table  at  the  end  of  the  volume.  For  calculating 
curves  it  is  not  necessary  to  use  more  than  four  decimals. 

A.  Bench  is  a  shoulder  hewn  with  the  axe  on  the  but- 
tressed base  of  a  tree,  and  so  shaped  at  the  top  as  to  afford 
footing  to  the  rod.  The  tree  is  blazed  and  the  elevation 
of  the  bench  marked  on  it  with  red  chalk.  Benches  serve 
as  permanent  reference  points  to  the  level.  They  are 
placed,  where  it  is  possible,  about  one  thousand  feet 
apart. 

Points.  The  operation  termed  pointing  is  the  fact  of 
putting  a  peg  firmly  into  the  ground,  and  of  driving  in  its 
top  a  tack,  or  making  thereon  an  indentation  whose  place 
is  indicated  by  cross  keel  marks,  directly  in  the  line  of  col- 
limation  of  the  transit.  Thus  true  lines  are  traced  on  the 
ground,  and  angles  measured  accurately.  When  the  transit 
is  set  over  a  point  it  is  so  posited  that  the  plumb  hangs 
immediately  above  the  tack  head.  If  the  head  plate  of 
the  tripod  be  much  inclined  the  plumb  should  be  examined 
after  levelling  the  instrument,  as  that  operation  disturbs  it 
to  some  extent. 

Stations.  The  line  of  a  survey  is  marked  on  the  ground 
at  regular  intervals,  by  stakes  two  feet  in  length,  blazed, 
and  numbered  from  0  up  in  arithmetical  progression. 
These  stakes  are  named  stations.  On  exploration  they 
are  commonly  placed  two  hundred,  and  on  location,  one 
hundred  feet  apart. 

It  is  customary,  when  locating,  to  drive  pegs  even  with 
the  surface  along  the  true  line,  and  to  place  the  stations  a 
couple  of  feet  to  the  right,  numbers  facing  in,  to  show  their 


EXPLANATIONS. 

position.  The  pegs  are  less  liable  to  disturbance  from  frost, 
animals,  &c. 

In  locating  for  construction  stakes  are  driven  on  sharp 
curves  at  intervals  of  50,  sometimes  25  feet. 

The  Chain  in  general  use  for  railway  surveys  is  made 
of  soft  iron.  It  is  100  feet  long,  and  divided  into  100 
links,  each  one  foot  in  length.  At  every  tenth  link  is 
attached  a  brass  drop,  toothed  so  as  to  indicate  its  distance 
from  the  end.  It  presents  the  advantages  of  durability, 
accuracy,  and  expedition. 


A  PRACTICAL  TREATISE 


ON 


RAILWAY  CURVES  AND   LOCATION. 


ARTICLE  I. 

OF    THE    INSTRUMENTS. 
THE   LEVEL. 

THE  level  is  an  instrument  used  in  ascertaining  the 
undulations  of  the  ground 
along  the  line  of  a  survey, 
and  of  measuring  these  ir- 
regularities accurately  in 
reference  to  an  assumed 
base  called  the  datum.  It 
consists  mainly  of  the 
telescope  k  i,  the  spirit- 
level  and  its  encasement 
6,  the  Y's  c  c,  the  rectan- 
gular bar  o  d,  the  axis  e,  the  plates  and  levelling  screws 
/  m,  and  the  tripod  g. 

In  the  tube  of  the  telescope  at  /*,  and  at  right  angles  to 
its  axis,  is  placed  a  flat  ring,  called  the  diaphragm.  To 
this  ring  the  cross-hairs  are  attached — two  delicate  spider 
lines  stretched  over  it  vertically  and  horizontally,  and  inter- 
secting at  the  centre.  It  isjield  jjrposition  by  four 


UNIVERSITY 


12  RAILWAY   CURVES   AND   LOCATION. 

slightly  movable  screws,  which  pierce  the  tube  in  the 
direction  of  the  "cross-hairs."  i  is  a  milled  head  for 
adjusting  the  focus  of  the  object  glass,  and  k  an  inserted 
tube,  containing  several  lenses,  which  may  be  moved  out 
or  in  so  as  to  make  the  spider-lines  distinctly  visible. 

A  straight  line  looked  along  from  the  eye  glass  at  k 
through  the  intersection  of  the  cross-hairs  is  the  line  of 
sight,  technically  named  the  line  of  collimation. 

The  immediate  supports  of  the  telescope  are  called  the 
Y's,  from  their  resemblance  to  that  letter.  If  a  small  arch 
were  sprung  between  the  two  legs  of  the  Y  it  would  give 
a  good  idea  of  the  clasping  pieces  which  hold  the  telescope 
in  place.  They  are  jointed  to  one  leg  and  secured  to  the 
other  by  pins  which  may  be  withdrawn  and  the  pieces 
turned  back  in  order  to  remove  the  telescope,  or  change  it 
end  for  end. 

The  Y's  are  attached  at  right  angles  to  the  bar  d,  which, 
again,  is  connected  firmly  at  right  angles  with  the  hollow 
axis  e.  This  latter  fits  closely  over  and  is  revolvable  hori- 
zontally around  a  solid  axis  s,  which,  passing  through  the 
plate/,  is  secured  to  the  head  of  the  tripod  by  means  of  a 
loose  ball-and-socket  joint.  The  plate /has  four  levelling 
screws  inserted  in  it ;  with  these  the  instrument  may  be 
brought  to  a  horizontal  position  even  when  the  lower  plate 
is  considerably  inclined. 

One  of  the  Y's  is  movable  for  a  short  space  up  or  down 
by  means  of  the  capstan-head  screw  o.  The  spirit-level  is 
likewise  movable  both  vertically  and  laterally  by  means  of 
screws  at  either  end. 

n  is  a  clamp  screw,  and  p  a  tangent-screw  for  slightly 
turning  the  telescope  in  a  horizontal  direction. 


THE   LEVEL  13 

To  adjust  the  Level. 

First.    To  make  the  line  of  collimation  coincide  with  the 
of  the  telescope. 

Set  the  instrument  firmly,  and  direct  the  telescope 
toward  some  distant,  distinct  object,  such  as  a  nail-head.  ^ 
Clamp  fast,  and  with  tangent-screw  fix  the  line  of  collima- 
tion upon  the  object  accurately.  Revolve  the  telescope 
half  way  round  in  the  Y's,  i.  e.  until  the  bubble  is  above 
it,  and  if  the  horizontal  spider-line  still  covers  the  point,  it 
requires  no  adjustment.  If  it  does  not,  reduce  the  error 
one-half  by  means  of  the  diaphragm  screws,  and  complete 
the  reduction  with  the  capstan-head  screw.  Revolve  the 
telescope  round  to  its  first  position,  and  if  the  horizontal 
line  and  point  do  not  then  coincide,  repeat  the  operation 
until  they  do,  in  any  position  of  the  telescope.  In  similar 
wise  the  vertical  hair  may  be  adjusted,  when  the  line  of 
collimation  should  cover  the  point  through  an  entire  revo- 
lution of  the  telescope. 

Great  care  should  be  taken  in  this  as  well  as  in  all  other 
adjustments  of  cross-hairs,  that  the  opposite  screw  of  the 
diaphragm  be  loosened  before  tightening  its  fellow,  or 
injury  to  the  instrument  must  result. 

Second.  To  make  the  axis  of  the  spirit-level  parallel  to 
the  line  of  collimation. 

With  levelling  screws  bring  the  bubble  to  the  middle  of 
its  tube,  reverse  the  telescope  in  its  Y's,  and  if  the  bubble 
does  not  then  stand  in  the  middle  correct  one-half  the 
deviation  with  the  screw  at  the  left  end  of  the  bubble-case, 
and  the  other  half  with  the  capstan-head  screw.  Again 
reverse  the  telescope  in  its  Y's,  and,  if  necessary,  repeat 
the  operation. 

Now  revolve  the  telescope  a  short  distance  in  its  Y's,  so 
as  to  bring  the  spirit-level  to  one  side  of  its  lowest  position. 
If  the  bubble  deviates  from  the  middle,  correct  the  error 

2 


14  RAILWAY   CURVES   AND   LOCATION. 

with  the  lateral  screws  at  the  right  end  of  the  bubble-case, 
and  examine  the  previous  adjustment  before  lifting  the 
instrument. 

Third.  To  bring  the  line  of  collimation  parallel  to  the 
bar. 

Turn  the  telescope  until  it  stands  directly  over  two  of 
the  levelling  screws,  and  with  them  bring  the  bubble  to  the 
middle  of  the  tube.  Then  revolve  the  telescope  horizontally 
until  it  stands  over  the  same  screws,  changed  end  for  end. 
If  the  bubble  does  not  still  stand  in  the  middle  of  the  tube, 
correct  one-half  the  deviation  with  the  capstan-head,  and 
one-half  with  the  levelling  screws. 

Place  the  telescope  over  the  other  levelling  screws  and 
proceed  in  a  similar  manner,  and  continue  the  corrections 
until  the  bubble  stands  without  varying  during  an  entire 
revolution  of  the  instrument  upon  its  axis. 

This  completes  the  adjustment  of  the  level. 

THE    ROD. 

The  rod  used  in  levelling  consists  of  a  staff  and  a  target, 
which  latter  is  so  attached  to  the  staff  as  to  be  movable 
along  it  from  end  to  end.  The  rod  is  commonly  seven  feet 
long,  but,  being  composed  of  two  rectangular  pieces  fitted 
together  by  means  of  a  sliding  groove,  it  can  be  extended 
to  nearly  double  that  length.  It  is  graduated  to  feet  and 
tenths  of  a  foot.  The  target  is  a  circle  of  wood  or  iron, 
usually  four-tenths  in  diameter,  and  divided  into  quadrantal 
sectors  by  a  horizontal  and  vertical  line  which  intersect  at 
its  centre.  The  sectors  are  painted  alternately  red  and 
white,  so  that  their  dividing  lines  are  visible  at  a  consider- 
able distance.  On  the  back  of  the  target,  where  it  meets 
the  graduated  side  of  the  rod,  is  fixed  a  chamfered  brass 
edging,  whereon  the  space  of  one-tenth  is  graven  from  the 
centre  down.  This  is  subdivided  into  ten  spaces  marking 


LEVELLING.  15 

hundredths,  and  these  latter  divided  into  halves,  so  that  the 
height  of  the  middle  of  the  target  above  the  base  of  the 
rod  may  be  accurately  read  to  within  -005  of  a  foot. 

There  is  a  similar  graduated  tenth  on  the  standing  part 
of  the  rod,  to  be  used  for  high  sights  when  the  sliding 
groove  comes  into  play. 

Both  target  and  rod  are  provided  with  clamp  screws. 

LEVELLING. 

The  operation  technically  called  levelling  is  performed 
thus : — 

Suppose  a  the  starting  point,  or  zero,  in  reference  to 
which  all  the  inequalities  of  the  surface  along  the  line  of 
survey  are  measured,  as  at  the  points  <?,  «,/.  The  hori- 
zontal line  af  is  called  the  datum  line.  This  is  arbitrarily 


assumed.  It  may  be  considered,  for  example,  at  any  dis 
tance  above  the  point  #,  and  the  irregularities  of  the  ground 
measured  from  an  imaginary  level  line  in  ether ;  but  for 
convenience  of  figuring,  and  other  politic  reasons,  it  is  cus- 
tomary in  seaport  towns  to  take  high  tide  as  datum.  In* 
land,  the  summer  surface  of  the  nearest  stream,  or,  when 
commencing  on  a  ridge,  the  highest  neighbouring  knoll  is 
issumed. 

Well !  suppose  a  to  be  zero,  and  the  instrument,  for 
nstance,  set  and  levelled  at  b.  Stand  the  rod  at  «,  and 
nilide  the  target  up  until  its  cross-lines  are  covered  by  the 
cross-hairs  in  the  telescope ;  i.  e.,  until  the  line  of  collima- 
tion  coincides  with  the  centre  of  the  target.  The  leveller 
directs  the  movements  of  the  target  by  raising  or  lowering 


16 


RAILWAY    CURVES    AND    LOCATION. 


his  hand.  A  circular  motion  of  the  hand  signifies  "  make 
fast."  The  bubble  should  always  be  examined  before  the 
rod  is  taken  down,  and  the  latter  should  be  read  twice,  or, 
if  convenient,  shown  to  the  leveller,  in  order  to  guard 
against  mistake.  If  in  this  case  it  reads  8  feet,  the  height 
of  the  instrument  is  then  8  feet  above  a.  To  find  the  ele- 
vation of  c  above  a,  take  the  rod  thither  and  lower  tho 
target  until  coincidence  results  as  before.  If  the  rod 
reads  2  feet,  of  course  c  is  8  —  2  =  6  feet  above  a. 

If  it  is  necessary  to  lift  the  instrument  here,  a  small  peg 
is  driven  at  c  before  sighting  to  that  point,  to  insure  firm 
footing  for  the  rod.  Sighting  back  from  the  new  position, 
d,  the  rod  reads  5  feet ;  then  5  -j-  t>,  the  elevation  of  c 
above  a,  =  11,  the  height  of  the  telescope  at  d  above  a. 
If  at  e  the  reading  is  6,  the  elevation  of  that  point  is  11 
—  6  =  5,  and  if  at  /  the  reading  is  8  the  elevation  of 
that  point  in  reference  to  a  is  11  —  8  =  3,  marked  -{-  3, 

The  rule,  therefore,  in  levelling  is,  at  each  new  stand  of 
the  instrument,  to  add  the  reading  of  the  rod  sighted  back 
at,  to  the  discovered  elevation  of  the  point  at  which  th<* 
rod  stands,  for  the  height  of  the  instrument ;  and  to  sub 
tract  from  this  height  the  reading  of  the  rod  at  any  points 
observed  from  the  new  position  in  order  to  find  the  eleva- 
tion of  those  points.  The  above  is  noted  in  the  field-bock 
as  follows: 


Station. 

Rod, 

Height  of 
Instrument. 

Total,  or 
Elevation. 

a 

8-00 

8-00 

00 

C 

2-00 

+  6.00 

5-00 

11-00 

e 

6-00 

-f-  5-00 

f 

8-00 

+  3-00 

i  .  . 

" 

OF   THE   TRANSIT.  17 

The  advantages  of  this  method  of  levelling  over  the  old 
system  of  backsights  and  foresights  are,  that  it  affords 
readier  facilities  for  testing  the  correctness  of  the  work, 
and  it  may  be  carried  on  more  rapidly.  By  the  old  plan 
each  sight  at  the  rod  was  linked  with  that  which  preceded 
it,  and  added  one  more  to  a  continuous  calculation  in  which 
a  single  error  affected  all  the  following  work.  Here,  how- 
ever, if  haste  is  required,  the  calculation  of  the  interme- 
diate sights  or  "cuttings"  may  be  omitted  entirely  while 
in  the  field,  the  reading  of  the  rod  only  being  set  down ; 
the  "totals"  may  be  worked  from  peg  to  peg,  and  the  lia- 
bilities to  mistake  thus  decreased  about  eighty  per  cent. 


OF   THE   TRANSIT. 

The  transit  is  an  instrument  for  measuring  horizontal 
angles.  It  consists  of  the  tele- 
scope a  c,  the  Y's  d,  the  compass- 
box,  &c.,  e  g,  and  the  axis  k. 
The  telescope  is  furnished  like 
that  of  the  level,  and  the  instru- 
ment is  similarly  fitted  to  its  tri- 
pod. The  telescope  revolves  in 
a  vertical  circle,  and  is  attached 
to  the  Y's  by  means  of  a  trans- 
verse axis  whose  extremities  turn  in  smooth  journals  at  the 
head  of  the  Y's.  The  body  of  the  instrument  at/  contains 
a  magnetic  needle,  with  its  usual  circular  surrounding, 
graduated  to  degrees  and  quarter  degrees.  The  flooring  of 
this  box  has,  on  one  side,  an  opening  with  chamfered  edge 
upon  which  the  vernier  is  engraved.  This  latter,  together 
with  the  telescope,  Y's,  and  all  the  upper  part  of  the 
instrument,  is  made  to  revolve  by  means  of  the  screw  h, 
upon  a  solid  plate  beneath,  which  is  likewise  graduated 
from  0  to  180°  each  way.  Thus  angles  may  be  measured 

2* 


18 


RAILWAY    CURVES    AND    LOCATION7. 


accurately  without  using  the  needle  at  all.  It  need  be 
regarded  merely  as  a  check,  g  is  a  clamp  screw  for  secur- 
ing the  plates  together,  and  i  a  screw  for  fastening  the 
needle  so  as  to  prevent  its  vibrations  while  the  instrument 
is  being  carried  from  place  to  place.  A  plumb  is  suspended 
from  the  axis  of  the  transit,  by  means  of  which  its  centre 
may  be  placed  over  a  point  on  the  ground. 


THE    VERNIER. 

The  vernier,  in  the  transit,  is  a  graduated  index  which 
serves  to  subdivide  the  divisions  of  the  graduated  arc  on 
the  lower  plate.  There  are  many  varieties  of  the  vernier, 
but  familiarity  with  one  renders  easy  the  acquaintance 
with  all,  since  the  same  general  principle  is  pervading. 


The  figure  represents  a  common  form.  Let  a  b  be  part 
of  any  graduated  arc,  and  c  d  the  vernier.  It  will  be 
observed  that  the  degrees  on  the  limb  are  divided  into 
spaces  of  15'  each.  Now  if  the  vernier  be  made  equal  in 
length  to  fourteen  of  those  spaces,  and  be  further  divided 
into  fifteen  equal  parts,  it  is  evident  that  each  of  these 
parts  will  contain  14'. 

Then,  if  0  of  the  vernier  coincides  with  any  division  of 
the  limb,  the  first  line  of  the  vernier  to  the  left  will  be 
just  one  minute  behind  the  first  line  of  the  limb  to  the 
left ;  the  second  vernier  line  two  minutes  behind  the  second 
limb  line,  and  so  on ;  so  that  if  the  vernier  be  moved  to 
the  left  over  the  space  of  15'  on  the  limb,  the  lines  from  0 
to  15  of  the  vernier  would  coincide  successively  with  lines 


TO    ADJUST    THE    TRANSIT  1J 

of  the  limb,  and  thus  any  angle  may  be  read  accurately  to 
minutes. 

The  vernier  in  the  figure  reads  48'  to  the  left.  A 
vernier  graduated  decimally  is  much  more  convenient  on 
railway  locations  than  those  with  the  common  graduation 
to  minutes.  This  is  principally  on  account  of  its  adapted- 
ness  to  running  in  curves  when  the  100  feet  chain  is  used. 
The  work  can  be  done  with  more  ease  and  rapidity.  One 
objection  to  it  is  that  the  tables  in  general  use  are  calcu- 
lated for  degrees  and  minutes. 

TO    ADJUST   THE    TRANSIT. 

Place  the  instrument  firmly  at  a,  level  it,  clamp  all 
fast,  and  with  tangent-screw  set  the  cross-hairs  on  the 
point  b,  at  any  convenient  distance.  Reverse  the  telescope 
on  its  axis,  and  fix  another  point  in  the  opposite  direction, 


..... 

of 


as  nearly  as  possible  equidistant  from  a.  Now  loose  the 
lower  clamp  and  revolve  the  entire  upper  part  of  the 
instrument  half  way  round  on  its  axis.  Clamp  fast,  and 
having  brought  the  cross-hairs  again  to  coincide  with  5, 
reverse  the  telescope.  If  the  sight  strikes  as  before,  the 
instrument  is  in  adjustment.  If  not,  place  another  point, 
d,  where  it  does  strike,  and  suppose  c  to  be  the  point  pre- 
viously fixed  :  the  point  e,  midway  between  d  and  c,  is  then 
in  the  straight  line.  With  the  adjusting  pin  carefully 
place  the  vertical  cross-hair  upon  /,  distant  from  d  one- 
quarter  of  the  space  d  c  —  with  tangent-screw  set  it  on  e. 
and  reverse  the  telescope.  If  the  points  have  been  cor- 
rectly placed,  and  the  hair  properly  moved,  the  sight  will 
strike  b,  and  the  adjustment  is  complete. 


20 


RAILWAY    CURVES    AND    LOCATION. 


After  finishing  this  adjustment,  the  telescope  may  still 
net  revolve  truly  in  the  meridian.  This  inaccuracy  there 
is  no  method  of  removing  in  the  field.  It  should  be  sent 
tc  an  instrument-maker  for  repairs. 


ARTICLE  II. 


PRELIMINARY   PROPOSITIONS. 


1.  In  any  circle  the  angle  o  cf  at  the  centre,  subtended 
by  the  chord  o  /,  is  double  the  angle  o  af,  at  any  part  of 
the  circumference  on  the  same  side  of  the  chord. 


2    The  angle  fbe,  formed  by  any  chord  /  6,  with  a, 
tangent  at  either  extremity,  is  called  a  tangential  angle. 


AVOIDING    OBSTACLES.  21 

and  is  equal  to  half  the  angle  /  c  b  at  the  centre,  or  is 
equal  to  the  angle  fa  b  at  the  circumference. 
.  3.  The  exterior  angle  dbf,  formed  at  the  circumference 
by  the  two  equal  chords  a  5,  b  /,  is  called  a  deflexion  angle, 
and  is  equal  to  the  central  angle  b  of,  or  double  the  tan- 
gential angle  e  bf.  df  is  called  the  deflexion  distance, 
and  e  /the  tangential  distance. 

4.  The  exterior  angle  p  o  m  of  two  unequal  chords,  is 
equal  to  the  sum  of  their  tangential  angles,  or  half  the  sum 
of  their  central  angles. 

5.  The  exterior  angle  i  k  o,  formed  by  tangents,  is  equal 
to  the  central  angle  b  c  o,  subtended  by  the  chord  which 
connects  their  points  of  contact  with  the  curve. 


ARTICLE  III. 

TO   AVOID   AN   OBSTACLE   IN   THE   LINE   OF   TANGENT. 

A  GLANCE  at  the  figure  will  show,  that  having  deflected 
to  <?,  and  placed  the  instrument  at  that  point,  the  angle 
he  d  must  be  made  equal  to  twice  d  be,  and  the  distance 


c  d  equal  to  the  distance  b  c.  Still  another  deflection  at 
d,  equal  to  the  original  angle  turned,  is  necessary  in  order 
to  sight  again  along  the  tangent. 

Should  the   obstruction  be  continuous  a  parallel  line 
may  be  run,  as  from  c  to  /,  by  deflecting  at  c  an  angle 


22  RAILWAY    CURVES   AND   LOCATION. 

equal  to  that  at  5,  and  at  /,  repeating  the  deflection  in 
order  to  strike  tangent. 

If  the  angle  dbc  exceeds  4°,  and  the  distance  be  is 
greater  than  200  feet, — or  even  with  an  angle  of  2|°, 
should  the  distance  be  greater  than  300  feet, — be  will 
differ  sensibly  in  length  from  b  k,  and  a  calculation  of  the 
latter  becomes  necessary.  To  effect  this,  multiply  the 
natural  cosine  of  the  angle  k  b  c  by  b  c.  This  result 
doubled  will  give  b  k  d,  the  length  proper  along  tangent. 

Thus : — Suppose  k  b  c,  the  angle  deflected,  to  have  been 
5°,  and  the  distance  b  c  340  feet.  Then  -9962,  the  natural 
cosine  of  5°,  multiplied  by  340,  gives  338*7  for  the  dis- 
tance b  k.  Double  this  makes  bkd  =  677-4,  and  shows 
a  difference  of  2-6  feet  between  b d  and  bed. 


ARTICLE  IV. 

SHOULD   THE    OBSTRUCTION   LIE    ON   THE   OPPOSITE   BANK 


AND  it  is  desirable  on  any  account  not  to  run  the  line 
from  dy  corresponding  to  a  d,  set  the 
instrument  at  a,  in  tangent,  and  deflect 
clear  of  the  obstacle  to  d.  Point  at  d. 
deflect  to  e,  and  point  also  there — 
marking  the  angles  c  a  d,  d  a  e.  Chain 
the  base  de,  and  placing  the  transit  at 
e,  measure  the  angle  dea.  Data  are 
thus  obtained  sufficient  for  the  calcula- 
tion of  the  line  d  a.  The  object  now  is 
to  find  the  point  c  and  the  angle  dca. 
The  angle  a  d  e  subtracted  from  180°  will  supply  the 


AVOIDING    OBSTACLES.  23 

angle  c  d  a,  so  that  in  the  smaller  triangle  we  have 
obtained  two  angles  and  their  included  side.  The  dis- 
tance cd,  and  angle  dca  readily  follow.  The  transit 
standing  at  e,  c  is  placed,  of  course,  in  the  prolongation 
of  the  base  d  e,  and  the  distance  c  d  is  carefully  set  off 
with  the  rod.  Moving  the  instrument  to  c,  and  turning 
the  angle  ecf  =  180°  —  dca,  we  are  again  in  tangent. 

Example.— Let  cad=  6°,  dae  =  35°,  dea  =  42°, 
and  the  base  d e  =  200  feet.  Then  in  the  triangle  dea 
we  have 

Nat.  sine  d  a  e  =  (-5736)  :  nat.  sine  d  e  a  =  (-6691)  : : 
d  e  =  (200)  :  d  «, 

•f)691  v  200 
Wherefore,     d  a  = &£>--  =  233-3  feet. 

*O  (  o\) 

Again,  the  angle  cda  =  77°.  The  angle  d  a  c  being 
— •  6°,acd  is  consequently  =  97°,  and  in  the  small  triangle 
we  have,  Nat.  sin.  a  c  d  —  (-9925)  :  nat.  sin.  c  a  d  = 
(•1045)::  da  =  (233-3):  c  d. 

•1045  X  233-3 
Therefore  c  d  =  -  —79925 =  24-564  feet,  and  d  cf 

180°  — 97°  =  83°. 

NOTE. — A  common  and  convenient  plan  for  triangulat- 
ing a  creek  is  as  per  figure.  Set  the  in- 
strument at  b,  fix  a  point  d  on  the  oppo- 
site shore,  and  making  d  b  c  a  right  angle, 
place  c  at  any  convenient  distance.  Now 
move  to  <?,  sight  to  c?,  and  making  d  c  a  a 
right  angle  also,  fix  a,  in  the  same  line 
with  b  and  d.  a,  c,  and  d  are  points  in 
the  circumference  of  a  circle  whose  dia- 
meter is  a  d,  a  b  :  b  c  : :  b  c  :  b  d,  and  therefore  Id  =  ~r 


24 


RAILWAY    CURVES    AND    LOCATION. 


ARTICLE  V. 

HAVING  GIVEN  THE  ANGLE  edb,  FORMED  BY  THE  INTERSEC- 
TION OF  TWO  STRAIGHT  LINES,  IT  IS  REQUIRED  TO  FIND 
THE  POINT  a  OR  6,  AT  WHICH  TO  COMMENCE  A  CURVE  OF 
<HVEN  RADIUS. 

Draw  the  bisecting  line  d  c.     Then  the  angle  d  c  i  = 

half  the  angle  acb  or  its 
equal  e  d  b  ;  and  in  the  tri- 
angle d  c  a,  the  angle  d  a  c 
being  a  right  angle,  we  have 
Rad.  of  1  :  Nat.  Tang. 
do  a  :  Rad.  ca :  ad.  There- 
fore ad  =  Nat.  Tang,  d  c  a 
X  Rad.  e  a. 

Example  1. — Let  e  d  b 
=  48°  and  a  c  =  1460 
feet.  Here  half  the  angle 
edb  or  acb  =  24°,  the 
Nat.  Tang,  of  which  is  *4452 ;  and  multiplying  by  Rad. 
1460,  we  have  650  feet  for  the  length  of  a  d  or  d  bj  the 
tangents. 

2. — If  a  die  given  and  radius  required, — 

ad  650 

Kad.  =  c^ ™ T—    A  A  /-->  —  1460. 

Nat.  Tang,  a  c  d        -4452 

The  following  rule?  are  approximate,  and  sufficiently 
correct  for  all  purposes  of  location. 

To  find  the  degree  of  curvature  of  a  b  divide  5730  by 
the  radius  in  feet ;  and  to  find  the  length  of  the  curve  in 
feet  divide  the  angle  a  c  b  (after  reducing  minutes  to  hun- 
dredths)  by  the  degree  of  curvature — the  chord  in  each 
case  being  100  feet  in  length. 


TO   TRACE   A   CURVE.  25 


ARTICLE  VI. 

TO    TRACE  A   CURVE  WITH   TRANSIT  AND   CHAIN.  t 

THE  legree  of  curvature  and  the  angle  to  be  turned  are 
known.  If  the  latter  is  expressed  in  degrees  and  minutes, 
reduce  the  minutes  to  hundredths,  since  the  100  feet  chain 
is  used,  and  divide  the  whole  angle  by  the  degree  of  cur- 
vature. The  quotient  will  be  the  length  of  the  curve  in 
feet,  and  the  P.  T.  is  at  once  ascertained. 


Let  m  a  be  the  tangent,  and  a  the  P.  C.  Place  the 
transit  at  a,  index  reading  0,  and  direct  the  sight  along 
the  tangent  m  a  e.  The  first  deflection  will  be  half  the 
central  angle  subtended  by  the  chord  used,  and  all  the 
stakes  put  in  from  a  will  be  fixed  by  similar  tangential 
deflections.  (Prelim.  Prop.  1.) 

3 


26  RAILWAY   CURVES   AND   LOCATION. 

When  the  point  d  is  reached,  the  angle  dab,  shown 
on  the  index,  will* be  half  the  angle  dbe,  or  its  equal 
a  c  d,  at  the  centre.  Move  the  instrument  to  d,  sight 
back  to  a,  and  turn  to  double  the  index  angle.  The 
telescope  is  now  directed  along  the  tangent  b  d  g,  and 
the  angle  dbe  =  acd  =  dab-{-adb1  reads  on  the 
index.  Note  this  angle  in  the  column  of  tangents  oppo- 
site station  d.  Continue  the  curve  from  this  new  posi- 
tion, precisely  as  was  done  at  a,  and  set  the  point  h. 
Move  to  h,  see  that  the  vernier  has  not  been  disturbed, 
and  sight  back  to  d.  The  index  now  shows  the  angle 
(d  b  e  -j-  h  d  </),  and  the  object  is  to  turn  the  angle  d  hf, 
i.  o.  repeat  the  angle  fd  A,  as  was  done  before  at  d,  and, 
at  the  same  time,  have  the  whole  angle  (d  b  e  -f-  hfy) 
indicated  on  the  plate.  To  effect  this,  merely  add  this 
angle  fhd  to  the  present  reading.  It  will  be  found  sim- 
pler, in  practice,  to  double  the  entire  angle  thus  far  turned, 
and  subtract  from  the  product  the  last  tangent,  viz.  d  b  e. 
The  vernier,  turned  to  this  resultant  angle,  will  put  the 
telescope  in  tangent  line  to  h.  And  so  on. 

^Example. — u  At  sta.  24  -f-  50  commence  a  4°  curve  to 
the  left  for  35°  12'."  Suppose  this  a  required  duty.  First, 
reducing  minutes  to  hundredths,  we  have  35° -20,  which, 
divided  by  4°,  gives  880  feet  for  the  length  of  the  curve. 
Adding  880  to  24  -j-  50  it  is  at  once  seen  that  sta.  33  +  30 
is  the  P.  T 

Let  a  be  the  P.  C.,  =  24  -f  50.  Now  the  deflexion 
angle  being  4°,  the  tangential  angle  is  2°,  with  a  chord  of 
100  feet.  With  a  chord  of  50  feet,  therefore,  the  tan- 
gential angle  is  1°,  and  this  deflexion  from  tangent  m  a  e 
fixes  station  25.  A  deflexion  from  this  latter  point  of  2°, 
the  chord  being  100  feet,  fixes  station  26.  And  so  on. 

When  you  have  fixed  the  point  d,  =  sta.  28,  the  index 
reads  7°.  .Move  up  to  station  28,  sight  back  to  the  P.  C., 
and  turn  the  index  to  14°.  This  throws  you  on  tangent 


TO    TRACE   A   CURVE.  27 

Proceed  as  before,  with  the  2°  deflexions,  to  sta.  31,  =  h. 
Move  up,  and  sight  back  to  sta.  28.  The  index  now  reada 
20°.  Multiplying  by  2,  and  subtracting  the  last  tangent, 
we  have  the  reading  of  the  tangent  at  h  =  26° :  we  have 
turned  26°  of  the  curve.  Continue  as  before.  After 
putting  in  sta.  33,  to  find  the  deflexion  which  shall  fix  the 
P.  T.,  33  +  30,  say,  as  100  feet  :  30  feet  : :  2°  :  the  re- 
quired deflexion,  =  36'.  We  may  here  remark  the  great 
convenience  of  an  instrument  graduated  to  hundredth  of 
a  degree  instead  of  sixtieths.  In  the  present  example  it 
would  be  seen  immediately  that  the  tangential  angle  for 
100  feet  being  2°,  for  1  foot  it  would  be  2  hundredth*  of 
a  degree,  and  for  30  feet  it  would  be  60  hundredths. 

Well !  when  the  P.  T.,  =  33  -f  30,  is  fixed,  the  index 
reads  30°  36'.  Move  up,  see  that  the  vernier  has  not  been 
disturbed,  and  sight  back  to  sta.  31.  Now  twice  the  index 
reading,  minus  the  last  tangent,  =  61°  12'  —  26°,  = 
35°  12',  the  present  tangent,  which  is  the  final  tangent, 
which  finishes  the  curve. 

The  advantage  of  this  manner  of  running  a  curve  is  that 
the  instrument  shows  at  a  glance  the  work  done,  and  there- 
fore errors  may  be  detected  with  greater  facility.  By 
comparing  at  the  P.  T.  the  total  index  angle  with  the 
distance  run,  the  work  is  tested  at  once. 


28  RAILWAY   CURVES   AND   LOCATION. 

The  above  is  recorded  in  the  field  book  as  follows 


6 


03   «     . 


J3    O 
•    o>  -M  o 

•3  S  oi2 


S 


g 


o          S 

ctt          co 


r-»  I— •  <M  <M  CO  CO 


§O  O  O  O  O  O  O 
O  O  O  O  O  O  CO 


In  running  compound  and  reversed  curves  the  operation 
is  quite  as  simple  as  the  foregoing.  A  point  is  fixed  at 
the  P.  C.  C.,  or  P.  R.  0.,  and  turning  into  tangent  at 
that  point,  the  second  curve  is  traced  from  this  tangent, 
•without  regard  to  what  precedes.  In  reverse  curving,  it 
is  a  good  plan  to  adjust  the  index  in  such  manner  at  the 
P.  R.  C.,  that  when  we  turn  into  tangent  it  will  read  0. 


TO    TRIANGULATE   ON   A   CURVE. 


29 


This  saves  troublesome  work,  and  it  is  advisable  moreover 
to  show  in  the  field-book  the  contained  angle  of  each  curve, 
as  well  as  the  test  of  the  two  tangents  with  the  magnetic 
course. 


ARTICLE  VII. 

TO   TRIANGULATE   ON   A  CURVE. 

SET  the  transit  at  a,  and,  as  usual,  sight  back,  and  turn 
into  tangent.  Estimate  the  distance  to  the  farther  bank 
— do  it  liberally — and  make  a  deflexion  around  the  curve, 
corresponding  to  your  estimated  distance.  Fix  a  point  b 
in  this  line.  Measure  any  convenient  angle,  b  a  c,  and  sut 


the  point  c.  Move  to  6,  measure  the  base  b  <?,  the  angle 
a  b  cy  and,  before  lifting  the  instrument  calculate  the  line 
b  a.  If  the  angle  turned  from  tangent  to  d  exceeds  4°, 
and  the  distance  is  greater  than  200  feet,  the  chord  a  d 
must  also  be  calculated,  as  per  example,  and  the  difference 
between  this  and  b  a  will  be  the  distance  b  d  to  the  point 
d,  in  the  curve,  which  can  be  fixed  from  b. 

3* 


30  RAILWAY   CURVES    AND    LOCAUON- 

Should  b  fall  between  d  and  a  the  operation  is  analogous. 

Example. — Let  a  be  a  point  in  a  6°  curve.  Having  set 
the  transit,  and  turned  into  tangent,  the  distance  to  the 
farther  verge  is  estimated  400  feet.  The  tangential  angle 
for  100  feet  is  3°,  and  to  fix  d,  400  feet  distant,  is  conse- 
quently 12°.  Deflect  this  angle,  fix  a  point  in  line,  and 
complete  the  triangulation,  as  previously  illustrated  in 
Art.  IV.,  p.  22.  Suppose  a  b  found  equal  to  472  feet. 
Now  the  tangential  angle  to  d  =  half  the  central  angle, 
=  fea,  =  12°;  and  to  find  the  length  of  the  chord  a  dy 
we  have,  in  the  triangle  efa, 

Rad.  :  Sin.  a  ef : :  ea  :  af,  that  is 

Rad.  of  1  :  Nat.  Sin.  12°  : :  9554,  the  Rad.  of  the  6° 
curve  :  half  the  chord  required.  Wherefore  a  d  =  twice 
the  Nat.  Sin.  12°  X  955-4  =  -2079  X  2  X  955-4  == 
397-25  feet.  Subtracting  this  from  472,  we  have  the 
distance,  74*75  feet,  back  to  the  point  in  the  curve.  Move 
the  instrument  to  d,  set  the  index  at  12°,  sight  back  to  a, 
and  turning  to  24°,  the  telescope  is  in  tangent.  A  deflexion 
of  3°  will  fix  the  next  station. 

NOTE. — In  this  case,  if  preferred,  a  third  proportional 
might  be  formed  with  the  chord  of  crossing,  as  shown  in 
the  note  to  Art.  IV. 


TO   CHANGE    THE   ORIGIN    OF   A    CURVE. 


31 


ARTICLE  VIII. 


TO  CHANGE  THE  ORIGIN  OF  ANY  CURVE,  SO  THAT  IT  SHALL 
TERMINATE  IN  A  TANGENT  PARALLEL  TO  A  GIVEN  TAN- 
GENT. 

LET  df/be  the  located  curve,  terminating  in  a  tangent/^, 
and  the  nature  of  the  ground  requires  that  it  should  ter- 
minate in  the  tangent  e  i,  parallel  to  fk.  At  /,  the  tele- 
scope being  directed  along  the  tangent  fk,  turn  to  the 


right  an  angle  equal  to  the  central  angle  d  bf,  previously 
turned  to  the  left  on  the  curve.  This  will  direct  the  tele- 
scope along  efy  parallel  to  d  I.  Measure  <?/,  and  go  back- 
on  the  tangent,  d  I,  a  distance,  c  d,  equal  to  it.  The  curve, 
retraced  from  c  and  consuming  the  same  angle,  will  termi 
nate  tangentially  in  e  i.  An  example  in  this  case  is  not 
necessary. 


82 


RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  IX. 


TO  CHANGE  A  P.  C.  C.  SO  THAT  THE  SECOND  CURVE  SHALL 
TERMINATE  IN  A  TANGENT  PARALLEL  TO  A  GIVEN  TAN- 
GENT. 

LET  a  b  d  be  the  compound  curve,  located  and  terminat- 
ing in  the  tangent  d  h.  Continue  the  larger  curve  to  e, 
and  from  e,  with  radius  e  I  =  k  6,  describe  the  curve  «?/, 
terminating  tangentially  in  / g,  parallel  to  d  h.  From  c, 
the  centre  of  the  larger  curve,  let  fall  upon  fg  the  perpen- 


dicular eg,  and  fill  up  the  figure  as  above.  Call  the  radii 
respectively  R  and  r,  the  angle  b  k  d,  or  its  equal,  k  e  m, 
x,  and  the  angle  e  If,  or  its  equal,  lcn,y.  Let  the  dis- 
tance if,  or  h  g,  be  named  D.  Now  the  line  c  g  is  made 
up  of  the  lines  c  m  -f-  m  h  -{-  kg,  i.  e.,  eg  =  cosin.  x 
(  R  —  r)  -f  r  -f  D.  c  g  is  also  made  up  of  the  lines  c  n 
+  ng,  i.  e.,  eg  =  cosin.  y  (R  —  r)  +  r.  Therefore  cosin. 
x  (R  —  r)  _|_  r  -f_  D  =  cosin.  y  (R  —  r)  +  r,  and  reducing, 


TO    CHANGE   A?P.  C.  C.  33 

cosin.  x  (R  —  r)  4-  D 
cosm.  y  ==• T^ ;  so  that  the  distance  z/, 

or  Jig,  measured  rectangularly  between  the  two  tangents, 
being  added  to  the  nat.  cosin.  x9  will  give  the  nat.  cosin.  of 
the  angle  e  If,  to  be  turned  on  the  smaller  curve.  The 
angle  y,  subtracted  from  the  angle  x,  gives  of  course  the 
angle  b  c  e,  to  be  advanced  on  the  larger  curve  ;  or,  divid- 
ing this  angle  by  the  degree  of  curvature  of  a  b,  we  find 
the  distance  from  b  to  e  the  P.  C.  C.  proper. 

If  ef  be  the  second  curve  located,  and  the  tangent  to  be 
touched  lies  within,  it  is  evident  that  we  must  retreat  upon 
the  large  curve,  and,  by  subtracting  D  from  the  cosine  of 
the  angle  y,  we  obtain  the  cosine  of  the  angle  x. 

^Example. — Suppose  a  b  a  3°  curve  located,  and  com- 
pounding, at  b,  into  a  6°  curve,  which  latter  is  continued 
to  the  right  through  an  angle  of  42°.  At  the  P.  T.  we 
discover  that  the  proper  tangent  is  64  feet  to  the  left. 
We  must  throw  our  curve  out,  then — we  must  advance  on 
the  3°  curve  a  certain  distance.  How  to  find  this  distance : 
The  radius  of  a  3°  curve  =  1910 ;  the  radius  of  a  6°  curve 
=  955-4;  R  —  r,  therefore,  =  954-6.  The  nat.  cosin. 
42°  =  -7431.  Now,  by  the  formula  just  obtained,  we 
cos.  x  (R  —  r)  +  D  (-7431  X  954-6)  4-  64 

- i '_ — - 

,  »-j  \  r\r  *    rt      '  ~~~- 

(R  —  r)  9o4-6 

•8101  =  nat.  cosin.  35°  53'.  Subtracting  this  from  42*, 
we  have  6°  07',  the  angle  to  be  advanced  on  the  8°  curve ; 
or,  reducing  minutes  to  hundredths,  and  dividing  by  3°,  we 
find  204  feet,  the  distance  from  b  to  the  correct  P.  C.  G 


34 


RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  X. 

SHOULD   THE   SECOND    CURVE   BE   ONE   OF   LONGER   RADIUS 
THAN   THE   FIRST, 

OUR  illustration  takes  simpler  form,  and  the  application 
of  D  varies  vice  versa. 


See  figure,  analogous  to  that  of  the  previous  problem. 
Here  of,  eg  are  equal  and  parallel  radii;  fh  a  perpen- 
dicular connecting  them.  Draw  its  fellow,  c  n.  Then, 
nf  =  c  h,  and,  consequently,  h  g  =  o  n.  Again,  letting 
k  m  fall,  perpendicular  to  eg,  we  have  c  m  =  n  I,  and  o  I 
=  c  m  -j-  o  n',  i.  e.,  cosin.  y  (R  —  r)  =  cosin.  x  (R  —  r) 
-f-D- 

We  observe  that,  with  a  curve  of  this  nature,  in  order 
tc  throw  the  line  farther  out,  it  is  necessary  to  go  back, 
toward  b ;  or,  having  located  to  y,  if  the  object  tangent  lie 
within,  we  must  advance  toward  e. 

Example. — Suppose  a  b  a  5°  curve,  b  the  P.  C.  C.,  and 
bg  a  2°  curve.  Setting  the  instrument  at  g,  the  P.  T., 
and  turning  into  tangent,  we  find  that  we  are  a  distance 
Jig,  =  53  feet,  too  far  to  the  left.  The  first  question  is, 
what  angle  have  we  turned  on  the  second  curve.  Let  it 


TO   CHANGE   A   P.  C.  0.  35 

be  28°.  Now  we  know,  that,  in  order  to  strike  farther  to 
the  right,  we  must  advance  on  the  5°  curve.  Consequently, 
D  must  be  added  to  the  cosine  of  28°,  to  give  us  the  cosine 
of  the  proper  angle  for  the  2°  curve  ;  and  the  difference 
between  28°  and  this  newly  found  angle  will  be  the  angle 
we  are  to  advance  on  the  5°  curve.  Thus  :  —  the  rad.  of  a 
5°  curve  =  1146  feet,  that  of  a  2°  curve  =  2865  feet 
and  their  difference  =  1719  feet.  The  nat.  cos.  of  28' 


=  •8829.     Then  =.  9137,  =  nat. 


cos.  23°  58*.     This,  subtracted  from  28°,  leaves  4°  02',  — 
80  feet,  from  b  to  the  correct  P.  C.  C. 

Synopsis  of  the  preceding  formulce. 

Call  D  the  distance  between  tangents  as  before,  a  the 
angle  of  the  second  curve  located,  and  b  the  angle  of  the 
same  curve  to  be  substituted  for  it. 

FIRST,  when  the  second  curve  has  the  smaller  radius — 
Tangent   falling  within  the  point,  cosine   b  = 
cos,  a  (R  — -r) J-J) 

(R-rj" 

Tangent  falling  without  the  point,  cosine  b  = 
cos.  a  (R  —  r)  —  D 

SECOND,  when  the  second  curve  has  the  larger  radius — 
Tangent  falling  within  the   point,  cosine   b   = 
cos,  a  (R  —  r)  —  P 
(R  —  r) 

Tangent  falling  without  the  point,  cosine  b  = 
cos.  a  (R  —  r)  +  D 


Very  little  attention  will  familiarize  these  formulae,  and 
render  the  field  practice  easy. 


36 


RAILWAY   CURVES   AND    LOCATION. 


ARTICLE  XL 

HAVING  LOCATED  THE  COMPOUND  CURVE  a  b  d,  TERMINAT- 
ING IN  THE  TANGENT  df,  IT  IS  REQUIRED  TO  FIND  THE 
P.  C.  C.  5,  AT  WHICH  TO  COMMENCE  ANOTHER  CURVE  OF 
GIVEN  RADIUS,  WHICH  SHALL  ALSO  TERMINATE  TAN- 
GENTIALLY  IN  df. 

PLOT  the  curves  as  per  figure.  From  c  let  fall  c  g,  per- 
pendicular to  the  tangent,  df.  From  k  and  «,  the  lesser 
centres,  drop  Jc  m,  i  Z,  perpendicular  to  c  g.  Call  the  great 
radius  R,  the  smaller  radius  r,  and  the  intended  radius  of 


the  second  curve  /.  Likewise  name  h  k  d,  the  angle  of 
the  small  curve  located,  x,  and  b  i  e  the  angle  to  be  found 
for  the  proposed  curve,  y.  Now,  the  tangent  df,  and  the 
curve  a  b,  lying  unstirred,  the  line  c  g  is  an  unvarying  dis- 
tance, and  it  is  made  up  of  the  lines  c  m  -f  m  g,  i.  e.,  c  g 
=  (R  —  r)  cosin.  x  +  r.  It  also  consists  of  the  lines  c  I 


COMPOUND   CURVES  37 

-f-  lg,  i.  e.,  eg  =  (R  —  /)  cosin.  y  -}-  /,  and,  reducing, 

(R  —  r)  cosin.  x  -f-  r  —  / 
nat.  cosin.  y  = —^ ,r .     This,  there  • 

fore,  is  the  formula  by  means  of  which  we  can  ascertain 
the  point  #,  as  follows  : — 

^Example. — Imagine  a  2°  curve,  a  A,  compounding  into 
a  6°  curve,  h  d,  which  terminates  at  c?,  in  the  tangent  df. 
The  tangent  lies  well ;  the  curve  a  h  likewise ;  but  it  is 
desired  to  throw  the  line  to  the  left,  on  better  ground, 
between  d  and  A,  by  means  of  an  intercalary  4°  curve. 
We  wish,  then,  to  know  the  distance,  h  b,  back  to  the  new 
P.  C.  C. 

The  radius  of  a  2°  curve  =  2865  feet,  of  a  6°  curve  =r 
955-4  feet,  and  their  difference  (R  —  r)  =  1909-6.  The 
radius  of  a  4°  curve  =  1433,  and  the  difference  (R  —  /) 
is.  therefore,  1432  feet.  Let  h  k  d,  the  angle  turned  on 
the  6°  curve,  be  41°,  the  nat.  cos.  of  which  =  -7547. 

T,        (-7547  X  1909-6)  +  9554  —  1433 

Then,  * —    1432~  ~  =  '  ^ 

nat.  cosin.  47°  43'.  Subtracting  41°,  we  have  6°  43',  the 
angle  h  c  b.  Reducing  minutes  to  hundredths,  and  divid- 
ing by  2°,  we  find  336  feet  to  bo  the  distance  from  h  to  6. 
A  4°  curve  of  47°  43',  traced  from  this  latter  point,  will 
terminate  in  the  tangent  ej. 


RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  XII. 

IF  THE  LATTER  CURVES  HAVE  LARGER  RADII  THAN  THE 


THE  solution  retains  its  shape  and  simplicity. 

Draw  the  figure  as  above,  and,  for  the  sake  of  uniformity, 
name  the  radii  as  before.  The  curve  a  6,  and  tangent  df, 
being  constant,  the  distance  m  g,  or  d  w,  is  here  constant 
Call  it  A.  Now  A,  in  the  first  place,  is  equal  to  d  i  —  n  i, 
i.  e.,  =  r  —  (r  —  R)  cosin.  x\  and,  in  the  second  place, 


it  is  equal  to  g  c  —  m  <?,  =  /  —  (/  —  R)  cosin.  y ;  where 
fore  r  —  (r  —  R)  cosin.  x  =  /  —  (/  —  R)  cosin.  ^,  and 

(r  —  R)  cosin.  x  -}-  /  —  r 
consequently  cosin.  y  =  - — > '  ,  ,  __    -p  \ • 

Example. — Suppose  a  b  a  Tc  curve,  compounding,  at  £, 
into  a  5°  curve,  b  d,  which  latter  subtends  an  angle  of  38°, 
and  terminates  in  the  tangent  df.  We  wish  to  substitute 
a  terminal  2°  curve,  Jig,  and  to  know  the  position,  A,  of 
the  new  P.  C  C. 

The  radius  of  a  7°  curve  =  819  feet  =  R.     r  and  r', 


TO    CHANGE   A    P.  R.  C. 


39 


the  radii  respectively  of  5°  and  2°  curves,  are  equal  to 
1146,  and  2865  feet,  r  —  R,  therefore,  =  327,  and 
/  —  R  =  2046  feet.  The  nat.  cosin.  of  38°  =  -788. 

mi         k    ^    r        i      ('788  X  327)  +  2865  —  1146 
Then,   by  the  formula,  *  —  —  on/ia  -    ~  == 


•9661,  =  the  nat.  cosin.  of  14°  57',  the  angle  to  be  turned 
on  the  2°  curve.  Subtracting  this  from  38°,  we  have  23° 
03',  the  angle  to  be  continued  on  the  7°  curve.  Reducing 
minutes  to  hundredths,  and  dividing  by  the  degree  of  curva- 
ture, 7°,  we  find  329  feet,  the  distance  from  b  to  the  new 
P.  C.  C.  h. 


ARTICLE  XIII. 


TO  CHANGE  A  P.  R.  C.,  SO  THAT  THE  SECOND  CURVE  SHALL 
TERMINATE  IN  A  TANGENT  PARALLEL  TO  A  GIVEN  TAN- 
GENT. 

• 

LET  a  d  I  be  the  reverse  curve,  located  and  terminating 


in  the  tangent  I  h.     Call  the  radius  c  J,  R,  and  the  radius 
b  e,  r.     Suppose   ig  the  given  tangent.     At  a  distance 


40  RAILWAY   CURVES   AND   LOCATION. 

from  it  equal  to  i  e,  the  radius  of  the  second  curve,  draw 
the  parallel  line,  op.  With  c  as  a  centre,  and  radius  of, 
=  R  -j-  r,  describe  the  integral  curve, /e,  cutting  op  in  e. 
d  then,  is  the  centre  of  the  curve  adjusted. 


Application. 

Place  the  transit  at  the  P.  T.,  Z,  and  turn  into  a  tan- 
gent, Im,  parallel  to  d  n,  the  common  tangent  of  the  two 
curves  at  d.  Unless  some  wide  mistake  has  been  made, 
the  distance  Z&,  measured  along  this  line  to  ig,  the  tan- 
gent proper,  will  be  about  equal  to  the  distance  ef,  and 
we  shall  have  the  proportion,  cf  :fe  : :  c  d  :  db,  i.  e.,  R 

+  r  :  ef  : :  R  :  d  b,  which  gives  -^5—   — ,   as  a  simple 

formula  for  finding  the  distance  back  from  d  to  5,  the  cor- 
rect P.  R.  C.  This  rule,  though  sufficiently  true  for  most 
cases,  is  not  mathematically  justifiable.  It  will  be  seen 
that  ef,  or  its  equal  i  I,  the  distance  we  wish  to  measure, 
is  a  curving  distance,  part  of  the  circumference  of  a -circle 
concentric  with  a  b.  Its  radius  is  (R  -f-  r),  therefore  its 

5730* 
degree  of  curvature  =  ,^         .,  or,  more   simply,  equals 

the  product  of  the  degrees  of  curvature  of  the  curves  com- 
posing the  reverse,  divided  by  their  sum.  To  be  strictly 
accurate,  then,  set  the  instrument  at  Z,  turn  into  tangent 
I  m  as  before,  and  trace  the  curve  i  Z,  until  it  strikes  the 
tangent  ig.  The  angle  which  il  subtends,  being  divided  by 
the  degree  of  curvature  of  a  6,  will  give  the  distance,  d  6, 
to  the  P.  R.  C.  proper.  The  curve  retraced  from  6,  will 
terminate  tangentially  in  ig,  and  its  angle,  bei,  will  be 
equal  to  df  I  —  deb. 

Example. — Let  a  d  I  be  a  reverse  curve,  composed  of  a 
3°  curve,  a  d,  and  a  6°  curve,  d  1.  Let  the  angle  df  I  be 
equal  to  52°,  and  suppose  the  distance  If  to  have  been 


WHEN   P.  C.    IS   INACCESSIBLE. 


41 


found  34  feet.  Being  part  of  a  2°  curve,  it  therefore  sub- 
tends a  central  angle  of  41'.  This  corresponds  to  a  dis- 
tance of  23  feet,  to  be  gone  back  on  the  3°  curve,  and 
52°-00  —  41'  =  51°  19',  the  angle  to  be  turned  from  h 
on  the  6°  curve,  in  order  to  strike  the  tangent  ig. 


ARTICLE  XIV. 

HOW   TO    PROCEED   WHEN    THE    P.  C.    IS   INACCESSIBLE. 

IN  the  figure,  drawn  to  illustrate  this  case,  let  c  be  the 
point  of  curvature,  c  a  the  tangent,  and  eke  the  curve. 
Now  the  angle  d  c  e,  included  between  the  tangent  and 


any  chord,  as  c  e,  fixing  the  point  e,  is  known.  Make  c  b 
along  tangent,  equal  to  c  e,  and  connect  be.  If  a  circle 
were  now  described  from  c  as  a  centre,  with  radius  c  e  or 
c  b,  d,  e,  and  5,  would  be  points  injiajurcjimference,  and 


42  RAILWAY   CURVES   AND   LOCATION. 

the  angle  dbe  at  once  proven  equal  to  half  the  angle  dot. 
With  proof  precisely  similar,  d  c  e  =  half  of  d  g  e,  and, 
consequently,  d  b  e  is  equal  to  one-fourth  of  the  central 
angle  subtended  by  the  chord  c  e. 

Example. — Suppose  c  to  be  the  inaccessible  point  of 
curvature  of  a  6°  curve,  eke.  It  is  concluded  to  run  to 
the  third  station,  e.  First  we  must  calculate  the  length 
of  the  chord  c  e.  The  angle  d  c  e  =  9°,  and  from  Art. 
VII.  we  have 

Rad.  of  1  :  9554  : :  nat.  sin.  9°  =  -1564  :  ^,  whereby 

e  G  is  shown  equal  to  298-8.  Place  the  transit  then  at  6, 
298-8  feet  distant  from  the  P.  C.,  and  deflect  to  the  left 
an  angle  of  4°  30',  equal  to  half  the  angle  dee.  This  is 
in  line  to  e,  and  b  e  must  likewise  be  calculated  as  follows : 

In  the  triangle  b  c  h  we  have 

Rad.  of  1 :  nat.  cosin.  4°  30'  =  -9969  : :  b  c  =  298-8  :  b  h 

b  e 
—  — ,  whereby  b  e  is  shown  equal  to  595-7  feet.    Arriving 

at  e,  the  index  reads  4°  30'.  Sight  back  to  6,  turn  to  18°, 
and  the  telescope  will  be  in  tangent.  Suppose,  however, 
that  having  reached  /,  100  feet  from  e,  this  latter  point  is 
also  found  inaccessible.  We  find  k  a  different  point  in  the 
curve,  thus:— The  angle  feg  =  18°  —  4°  30'  =  13°  30', 
and  the  tangential  angle  g  e  k  =  3°.  Consequently  the 
angle  fek  =  10°  30',  and,  drawing  the  bisecting  line  e  i, 
we  have,  in  the  triangle  efi, 

Rad.  of  1  :  nat.  sin.  5°  15'  =  -0915  : :  ef  =  100  :  fi 
=  9-159  feet.  Therefore  fk  =  18-318  feet,  and  the 
angle  efk  =  90°  —  5°  15'  =  84°  45'.  At  /  deflect  this 
angle  to  the  right,  and  measure  the  distance  fk  carefully 
*vith  the  rod.  At  k,  sighting  back  to  /,  and  turning  the 
equal  angle  fke,  the  telescope  will  be  directed  to  e,  and 
the  curve  may  be  continued. 

If  it  is  inconvenient  to  run  the  line  b  e,  the  point  e  may 


TO   AVOID    OBSTACLES    IN    THE   LINE    OF    CURVE.  43 

be  reached  thus : — Fix  the  P.  C.     Find   the   tangential 
distance  d  e,  corresponding  to  the  angle  dee.     Carefully 
with  the  rod  lay  off  b  I,  equal  to  it,  at  right  angles  to  b  e 
Set  the  transit  at  Z,  and,  in  line  with  c,  put  in  e. 
The  distance  b  I  should  not  exceed  10  or  12  feet. 

NOTE. — The  foregoing  illustrations  will  apply  when  the 
P.  T.  is  likewise  inaccessible. 


ARTICLE  XV. 

TO  AVOID   OBSTACLES   IN  THE   LINE   0"F  CURVE. 

LET  b  Jc  h  be  the  curve.  We  can  either  follow  the  tan- 
gents h  d,  db,  or  trace  a  parallel  curve,  g  a,  within  the 
first ;  which  tracing  is  effected  thus  : — Set  the  instrument 


at  A,  the  P.  C.,  and  offset  any  distance  Jig,  at  right  angles 
to  the  tangent  h  d.  It  will  be  observed  that  as  the  dis- 
tance h  g  increases,  the  distance  g  a  decreases,  whilst  the 
angle  subtended  by  g  a  remains  equal  to  that  subtended 


44  RAILWAY   CURVES   AND   LOCATION. 

by  h  b ;  i.  e.,  our  deflexions  on  the  offset  curve  stand 
unchanged,  but  the  corresponding  chords,  g  /,  f.e,  &c., 
are  less  than  their  equivalents,  h  i9  i  k,  &c.,  along  h  b. 
To  find  their  length,  h  i,  i  k,  &c.,  being  equal  to  100 
feet,  we  have  the  proportion,  c  h  :  eg  ::  hi:  x;  i.  e., 
R,  :  rad.  —  h  g  : :  100  feet  :  x,  where  x  symbols  the  un- 
known chord.  Now  set  the  transit  at  g,  turn  into  tangent 
parallel  to  h  d,  and  with  the  shortened  chord,  fix  fea. 
Rectangularly  to  the  tangents  at  these  points,  and  distant 
h  g,  will  be  i,  k,  b  of  the  curve  proper. 

Example. — Let  b  h  be  a  4°  curve,  and  the  offset  distance 
85  feet.  The  radius  then  is  1433,  and 

1433  :  1348  : :  100  :  94,  the  short  chord. 

To  follow  the  tangents,  suppose  the  angle  b  c  h  =  42°. 
Then  by  Art.  V.  we  find  the  tangent  h  d  =  550  feet, 
which  distance  we  duly  measure,  and,  at  d,  deflecting  42°, 
lay  oft  an  equal  distance  to  b,  the  point  of  tangency. 


FIND   THE   RADII   OF   REVERSE   CURVE. 


45 


ARTICLE  XVI. 

HAVING  GIVEN  THE  ANGLES  d  b  k,  mkl,  AND  THE  DISTANCE 
b  &,  IT  IS  REQUIRED  TO  FIND  THE  RADII  C  6,  ef  OF  THE 
EASIEST  REVERSE  CURVE  WHICH  SHALL  UNITE  a  d,  ~k 

THE  angle  d  b  e  is  equal  to  the  angle  a  c  e,  half  of  which 
is  b  c  e.     So  likewise  ef  k  is  equal  to  half  of  I  km. 


Then,  [nat.  tang,  b  c  e  -f  nat.  tang,  efk]  :  nat.  tang, 

b  c  e  : :  b  k  :  b  e,  and  b  k  —  b  e  =  e  k.    Wherefore  rad.  c  e 

be  ek 

=  — v —  and  rad.  ef  = 7— ?-. 

nat.  tang,  bee  nat.  tang,  kfe 

Example. — Suppose  the  angle  d  b  e  =  54°  30',  the 
angle  lkg  =  W  20',  and  the  distance  Ik  =  832  feet. 

Therefore  the  angle  b  c  e  =  27°  15',  the  nat.  tang,  of 
which  is  -5150,  and  the  angle  efk  =  16°  40',  the  nat. 
tang,  of  which  is  -2994.  The  sum  of  the  tangents  =  -8144. 
Then,  to  find  b  e,  we  have 

As  -8144  :  -5150  : :  832  :  526,  and  subtracting  this  from 
b  k,  we  have  e  k  =  306  feet. 

526  306 

Again,  the  radius  c  e  =  ^TH,  and  the  radius  ej  =  .2994? 

=  1022  feet. 


46  RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  XVII. 

HAVING  GIVEN  THE  CURVE  /(/,  LOCATED  AND  TERMINATING 
IN  THE  TANGENT  g  m,  IT  IS  REQUIRED  TO  FIND  WHERE 
A  CURVE  OF  DIFFERENT  RADIUS  WILL  TERMINATE  IN  A 
TANGENT  PARALLEL  TO  g  m. 


be  the  curve  located,  and  fh  the  curve  proposed, 
commencing  at  the  common  point  /,  and  terminating  in 
the  parallel  tangents,  gm,hl.  We  wish  to  find  the  length 
und  direction  of  the  line  g  h,  connecting  the  points  of  tan- 


Call  the  radii  respectively  R  and  r,  and  the  central 
angle,  fdg  orfch,  x.  Now  g  k  =  d  e,  =  sine  of  x,  c  d 
being  radius  ;  i.  e.,  =  nat.  sin.  a;  X  (R  —  r).  Again,  the 

£%• 

angle  kg  h  =  fh  b  =  -,  and  g  k  =  cosin.  kg  h,  g  h  being 

SY* 

radius,  i.  e.,  g  k  =  nat.  cos.  -  X  g  h,  and,  consequently. 

£• 

nat.  cos.  -  X  g  h  =  nat.  sin.  x  X  (R  —  r),  wherefore 

(R  —  r)  nat.  sin.  x 
gh=  — . 

nat.  cos.  <r 
It  will  be  observed  that  the  angle  leg  h,  included  between 


TERMINATION   OP   A   CURVE.  47 

the  tangent  to  a  curve  at  any  point  g,  and  the  line  g  h 
connecting  </  with  an  equivalent  point  h  in  any  other  curve 
fh  commencing  at  the  same  P.  C.,/,  and  turning  in  the 
same  direction,  is  invariably  equal  to  half  the  common 
central  angle,  fdg  orfch. 

Example.  —  Let  fg  be  a  7°  curve,  subtending  a  central 
angle,  fdg,  of  44°  26'.  Having  arrived  at  g,  the  P.  T., 
and  turned  into  tangent,  g  m,  it  is  desired  to  fix  the  P.  T., 
A,  of  a  4°  curve  which  shall  likewise  subtend  an  angle  of 
44°  26'.  Here  the  radii  are,  respectively,  819  and  1433 
feet,  and  E,  —  r  =  614.  The  nat.  sine  of  44°  26'  =  -700, 
and  the  nat.  cosine  of  half  this  angle,  viz.  :  22°  13'  = 

614  y  *7 
•9258.     Then,  by  the  formula,  g  h  =  ---  =  464-2. 


Deflecting,  therefore,  to  the  left,  an  angle  of  22°  13',  and 
laying  off  the  distance  464-2  feet,  we  arrive  at  the  point  h. 
Move  to  A,  sight  back  to  g,  and  a  deflexion  of  22°  13'  to 
the  right  will  direct  the  telescope  along  the  tangent  h  L 

If  h  were  the  P.  T.  located,  and  g  the  point  required, 
the  same  angle  and  distance  would  apply. 


48  RAILWAY   CURVES   AND   LOCATION. 


ARTICLE  XVIII. 

HAVING  THE  CURVE  nfg  LOCATED,  AND  TERMINATING  IN 
THE  TANGENT  g  W,  IT  IS  REQUIRED  TO  FIND  THE  POINT 
/,  WHEREAT  TO  COMPOUND  WITH  ANOTHER  CURVE  01 
GIVEN  RADIUS,  WHICH  SHALL  TERMINATE  IN  i  I,  PARAL- 
LEL TO  g  m. 

[See  previous  figure.] 

NAME  the  radii  and  angle  as  before.  Measure  the  dis- 
tance g  i,  between  the  tangents,  and  call  it  D.  Then  c  h 
is  equal  to  c  e  -j-  e  k  -f-  g  i  or  k  h ;  i.  e.,  R  =  (R  —  r) 

R_(r_D) 
cosm.  x  -f-  r  -\-  D,  or,  transposing,  cosm.  x  =  — j^ \ — • 

Thus  discovering  the  angle  f  d  g^  divide  it  by  the  curvature 
nfg,  and  we  have  the  distance,  <//,  to  the  P.  C.  C./.  The 
second  curve,  traced  from  this  point,  will  terminate  tan- 
gentially  in  h  I. 

^Example. — Let  the  curvatures  equal  those  of  the  last 
problem,  and  suppose  the  distance  g  i  to  be  175-6  feet. 

Then,  by  the  formula,  cosin.  x  =  —   — gTT~~     ~  ==  '7143 

=  cosin.  44°  26'.  Reducing  minutes  to  hundredths,  and 
dividing  by  7°,  we  have  635  feet,  the  distance  back  to  the 
P.  C.  C. 


COMMENCEMENT    OF   A    CLRVE 


49 


ARTICLE  XIX. 

HAVING  GIVEN  A  TANGENT  a  b,  AND  A  CURVE  b  k,  LOCATED, 
IT  IS  REQUIRED  TO  FIND  THE  POINT  d,  OR  /,  AT  WHICH 
TO  COMMENCE  A  CURVE  OF  GIVEN  RADIUS,  WHICH  SHALL 
BE  TANGENT  TO  BOTH. 

DRAW  the  radius  b  c,  and  call  it  R.  Name  the  radius 
of  the  other  curve  r.  Make  b  e  equal  to  r,  and,  through 
e.  draw  the  line  e  I,  parallel  to  the  tangent  a  b.  From  c 
as  a  centre,  with  eg,  =  (R  -|-  r),  as  radius,  sweep  the  arc 
of  a  circle ;  which  arc  will  intersect  #  I  at  g,  and,  from  the 
equidistances,  prove  g  the  centre  of  the  other  curve  touch- 
ing a  by  b  k,  tangentially,  at  the  points  d  and  /. 


Now,  to  find  the  point  /  for  purposes  of  location,  we 
must  know  the  angle  b  cf.  Call  it  x.  In  the  triangle 
eg  e,  we  have  eg  :  c  e  : :  radius  :  cosin.  g  c  e,  or  b cf\  i,  e*, 
(R  -f  r)  :  (R  —  r)  : :  1  :  nat.  cosin  #,  wherefore  nat.  cosin. 

x  =  TO" r — {•     So  that,  having  divided  the  difference  of 

^rt  -f-  r) 

the  radii  by  their  sum,  we  shall  find  opposite  to  the  quo- 
tient, in  the  table  of  nat.  cosines,  the  angle  b  cf  required. 
This  angle,  divided  by  the  degree  of  curvature  of  b  k,  will 

5 


60  RAILWAY   CURVES   AND    LOCATION 

give  the  distance  from  b  to  the  P.  R.  C.,  /;  and  180°  —  x 
will  equal  the  angle  f  g  d,  to  be  turned  on  the  other  curve. 

Another  plan,  which  may  be  preferred,  for  finding  the 
point  d,  is  as  follows  : 

Make  c  g  the  diameter  of  a  semicircle,  ctg.  This  semi- 
circle is  tangent  to  b  d  at  t,  and  tf,  perpendicular  to  eg, 
is  a  common  tangent  to  the  curves  b  k,  df.  We  have  then, 

«/  X  /^  =  tf\  i.  e.,  R  X  r  =  tang.2  1.    Multiplying  the 

radii  together,  and  extracting  the  square  root  of  the  product, 
we  find  the  distance  tf,  or  its  equal  t  b,  which,  doubled,  is 
the  distance  b  d,  from  b  to  the  point  of  curvature,  d. 

Example.  —  Suppose  k  b  a  3°  curve,  tangent  to  the  line 
a  b.  We  wish  to  know  the  point  /,  at  which  to  begin  a  5° 
curve,  which  shall  be  tangent  to  both.  Here  R  =  1910, 
and  r  =  1146,  wherefore  (R  +  r)  =  3056,  (R  —  r)  = 


764,  and       =     =  -       ,  =  -25,  =  nat.  cosh,  75°  31'. 

Reducing  minutes  to  hundredths,  and  dividing  by  3°,  we 
have  2517  feet,  the  distance  from  b  to  the  P.  R.  C.,/. 

If  the  point  d  be  required,  we  have  ^/1910  X  1146  = 
1478-9.  Doubling  this  result  we  find  2957'8  feet,  the  dis- 
tance from  b  to  the  P.  C.,  d. 

If  the  radii  are  equal,  of  course  the  distance  b  d  is  equal 
to  their  sum,  and  the  angle  #  is  a  right  angle. 


TO  RUN  A  TANGENT  TO  TWO  CURVES.         51 


ARTICLE  XX. 

TO  RUN  A  TANGENT  TO  TWO  CURVES. 

LET  g  £,  e  /r,  be  the  two  curves.  First  plot  them 
carefully  to  a  large  scale,  and,  finding  from  this  plot  an 
approximate  P.  T.,  say  6,  run  the  tangent  ef.  Now  the 
chord  in  the  curve  g  b,  to  which  ef  is  parallel,  is  known, 
and  consequently  the  shortest  distance,  fg,  between  the 
tangent  and  the  curve  may  be  measured.  Then  ef  :  f g 

::  rad.  :  tang,  feg,  and  nat.  tang,  feg  =  -^>  Practi- 
cally, this  angle  feg  will  be  found  equal  to  g  a  b  or  dc  e, 
so  that,  dividing  it  by  the  degree  of  curvature  of  k  <*,  we 
shall  have  the  distance,  e  d,  to  the  proper  P.  T. 


Strictly  speaking,  the  angle  g  ef  is  too  great  by  the 
small  angle  bhg.  There  are  two  modes  of  calculating 
this  latter,  but  they  are  both  complex,  and  in  any  but  a 
most  unusual  case,  the  above  rule  is  sufficiently  accurate. 

Example. — Let  k  e  be  a  5°  curve.  Suppose  the  tangent 
ef  =  1632  feet,  and  the  distance  gf  =  42  feet, 

Then         9  =  -0257  =  nat.  tang,  of  1°  28'.     Reducing 


52 


RAILWAY   CURVES   AND   LOCATION. 


minutes  to  hundredths,  and  dividing  by  5 
feet,  the  distance  back  to  the  correct  P.  T. 


we  have  29 


Should  the  curves  turn  in  the  same  direction,  or  should 
e  be  a  fixed  point  from  which  to  run  a  tangent  to  g  b,  the 
above  illustration  is  still  applicable. 


ARTICLE  XXI. 

OBDINATES. 


TO   FIND   THE   MIDDLE   ORDINATE  TO  ANY  GIVEN  CHORD,  IN 
A  CURVE  OF  ANY  GIVEN  RADIUS. 


1st.  (See  figure  in  Art.  XVL)  Ic  c  =  */  a  e2  —  a  F,  and 
a  c  or  be  —  k  e  =  the  ordinate  required.  That  is,  from 
the  square  of  the  radius  subtract  the  square  of  half  the 
chord,  and  take  the  square  root  of  the  remainder  from 
radius,  for  the  middle  ordinate. 


Example. — The  radius  a  c  being  819  feet,  and  the  chord 
a/ 100  feet,  to  find  the  middle  ordinate,  Ik. 

Here  a  <?  —  a  t?  =  670761  —  2500,  =  668261,  the 
square  root  of  which  is  c  k,  =  817-5,  which  taken  from 
radius  819,  leaves  1-5,  the  required  middle  ordinate. 


ORT>I  NATES.  63 

2d.  Subtract  the  nat.  cosine  of  the  tangential  ar/gle 
from  1,  and  multiply  the  remainder  by  radius. 

Example. — Suppose  a  b  a  7°  curve.  Here  the  nat. 
cosine  of  3°  30',  the  tangential  angle,  is  -9981,  which,  sub- 
tracted from  1,  leaves  *0019.  Multiplying  this  latter  by 
radius  819,  we  have  1*5,  the  middle  ordinate  as  before 


HAVING  GIVEN  THE  MIDDLE  ORDINATE,  TO  FIND  ANY  OTHER. 


1st.  eg  =  \/  c  e2  —  eg*,  and  ed  =  eg  —  g  d  or  clt. 

Example. — Suppose  the  distance  Jed  or  eg  to  be  20 
feet.  Thence  e  e2  —  c  g2  =  670761  —  400  =  670361.  the 
square  root  of  which  is  eg,  =  818*76.  Taking  from  this 
the  distance  gd  =  817 '5,  we  have  the  ordinate  ed  = 
1-26. 

*2d.  Multiply  the  ordinates  of  a  1°  curve  by  the  deflexion 
angle  of  the  curve  whose  ordinates  are  required.  This  is 
an  approximation  sufficiently  exact  for  railwa-y  curves. 


54 


RAILWAY    CURVES   AND   LOCATION, 


ARTICLE  XXII. 


TO    FIND   THE   RADIUS    CORRESPONDING   TO  ANY  DEFLEXION 
ANGLE,  AND  TO  EQUAL  CHORDS  OF  ANY  GIVEN  LENGTH. 

HERE  the  deflexion  angle,  d  b  a,  is  of  course  equal  to 
the  central  angle  d  c  b,  subtended  by  the  given  chord  d  6, 

SIA-*'-       i       ,jiv  •      •                             180°  —  deb 
and  the  triangle  c  d  b  being  isosceles  we  have ~ — 

A 

=  c  d  b,  or  d  b  c. 

Then  nat.  sin.  deb  :  nat.  sin.  c  b  d  ::  db  :  c  d9  the 
radius. 


Example. — Let  the  deflexion  angle  be  5°,  and  the  chord 
100  feet.  Required  the  radius.  The  nat.  sine  of  5°  = 

-I  OAO CO 

•0872  and          0          =  87°  30',  the  nat.  sine  of  which  is 
z 

9990. 

Then  -0872  :  -9990  : :  100  :  radius  =  1146  feet. 

An  approximate  result,  sufficiently  accurate  for  all  pur- 
poses of  railway  location,  may  be  found  by  simply  dividing 
5730  by  the  deflexion  angle. 

To  find  the  deflexion  angle  corresponding  to  any  given 


RADIUS    CORRESPONDING    TO    DEFLEXION    ANGLE.  .> 

radius  and  chord. — In  this  case,  we  have  c  d  :  df :  :  rad 
of  1  :  nat.  sin.  of  half  the  deflexion  angle  ;  therefore,  divide 
half  of  the  chord  by  the  radius  of  the  curve  ;  the  quotient 
will  }yc  the  nat.  sine  of  half  the  deflexion  angle. 

JExample. — Radius  as  before  1146  feet,  and  chord  100 

feet.  Then  j||g  =  -0436,  the  nat.  sin.  of  2°  30'.  Dou- 
bling this  result,  we  have  the  deflexion  angle,  viz.  5°  00'. 

To  find  the  deflexion  distance  with  chord  of  100  feet  and 
any  radius. — Divide  the  constant  number  10000  by  the 
radius  in  feet ;  the  quotient  will  be  the  deflexion  distance  : 
— for  the  deflexion  distance  with  a  radius  of  10000  ft-et  is 
1  foot,  and  the  deflexion  distances  for  other  radii  increase 
inversely  as  the  radii. 

Example. — What  is  the  deflexion  distance  for  a  5°  curve, 

the  chord  being  100  feet?     Here  ^^  =  8-72  feet,  the 

deflexion  distance. 

To  find  the  deflexion  distance  with  any  given  chord  and 
radius. — The  deflexion  distance  is  equal  to  twice  the 
natural  sine  of  half  the  deflexion  angle,  multiplied  by  the 
chord.  Thus,  the  chord  being  100  feet,  and  the  deflexion 
angle  5°,  we  find  the  nat.  sine  of  2°  30'  equal  to  -0436, 
which  doubled,  and  multiplied  by  100,  gives  8*72  feet,  as 
above. 

The  tangential  distance,  with  any  radius  and  chord,  is 
in  like  manner  equal  to  twice  the  nat.  sine  of  half  the  tan- 
gential angle  multiplied  by  the  chord.  Thus,  the  tangential 
angle  being  2°  30'  and  the  chord  100  feet,  we  find  the  nat. 
S'ne  of  1°  15'  =  -0218.  Multiplying  by  2,  and  100,  we 
have  4*36  feet,  the  tangential  distance. 

For  all  curves  under  11°,  the  tangential  distance  is  equal 
to  half  the  deflexion  distance. 


J>6  RAILWAY   CURVER   AND   LOCATION 


ARTICLE  XXIII. 

OF   EXCAVATION   AND    EMBANKMENT. 

A  RAILWAY  line  having  been  located,  the  first  office  dutiea 
are  to  map  it  down,  to  make  a  continuous  profile,  and  an 
approximate  estimate  of  the  cost  of  grading,  &c.  The  "  ele- 
vation" of  each  stake  above  or  below  a  certain  assumed 
base  being  fixed,  the  profile  is  drawn,  on  paper  prepared 
for  the  purpose,  to  a  horizontal  scale  of  400  feet,  and  a 
vertical  scale  of  40  feet,  to  the  inch.  This  distorted  pic- 
ture presents  at  a  glance,  in  compact  shape,  the  undula- 
tions of  the  surface,  and  from  it  grades  are  adopted,  to 
balance  as  nearly  as  possible  the  excavation  and  embank- 
ment. By  these  grades,  noted  in  the  record,  the  cutting 
or  filling  is  ascertained  at  each  station.  Suppose,  for 
instance,  that  the  elevation  of  station  40  is  12  feet,  and 
that  of  station  100  is  72  feet.  The  distance  between  40 
and  100  is  6000  feet,  and  the  difference  between  12  and 
72  is  60  feet.  Consequently,  to  connect  those  points,  we 
require  an  ascending  grade  of  1  in  100  feet.  Grade  at 
station  54  is  therefore  26  feet,  and  at  station  60,  32  feet. 
If  the  elevation  at  station  60  be  38  feet,  of  course  we  have 
6  feet  of  excavation,  and  this  is  marked  in  our  estimate 
sheets  -f  6.  If  the  elevation  at  that  point  should  be  27 
feet,  o  feet  of  embankment  is  the  consequence,  which  is 
marked  accordingly,  —  5 ;  plus  indicating  excavation,  and 
minus  embankment. 

The  usual  slope  for  embankment  is  1£  feet  horizontal 
to  1  foot  vertical,  making  an  angle  of  about  34°  with  the 


OF   EXCAVATION    AND   EMBANKMENT.  57 

horizon ;  that  for  earth  cut,  1  to  1,  or  45°,  and  for  rock 
cut  J  to  1,  or  76°.  The  slopes  of  the  ground  surface  at 
each  station  being  known,  together  with  the  breadth  of  the 
intended  roadway,  we  are  prepared  to  calculate  the  cross- 
sectional  areas,  and  from  them  to  determine  the  cubic  yards 
of  excavation  and  embankment. 

To  facilitate  these  operations  the  following  rules  were 
prepared.  With  Trautwine's  common  diagram,  and  the 
table  of  squares  and  square  roots  appended  to  the  volume, 
they  will  be  found  an  observable  assistance. 


Suppose  li  the  road  bed,  and  fh  the  depth  of  an  ordi 
nary  clay  cut.  Produce  the  side  slopes  until  they  meet  in 
the  point  k.  Then  the  angle  being  45°,  ef=fk,  and 
ef2  =  twice  the  triangle  efk,  =  the  triangle  e  kg.  For 
like  reasons  I  h  =  h  k,  and  I  h2  =  triangle  I  k  i.  But  the 
triangle  Iki,  taken  from  the  triangle  ekg,  leaves  the  area 
elig,  to  find  which  we  have  therefore  the  following 

Rule. — To  half  the  breadth  of  the  roadway,  add  the 
depth  of  the  cut.  From  the  square  of  this  sum,  subtract 
the  square  of  half  the  breadth  of  the  roadway  for  the 
area. 

Example. — Suppose  the  breadth  of  the  roadway  to  be 
32  feet,  and  the  depth  hf,  7  feet.  Half  of  32  =  16,  the 
square  of  which,  viz.,  256,  becomes  a  constant  subtrahend. 
16  •-{-  7  =  23,  and  a  reference  to  the  table  shows  the 
square  of  23  equal  to  529.  Therefore  529  —  256  =  273, 
the  area  required  in  square  feet. 

If  there  be  a  regular  slope,  as  eg  (p.  58),  we  can  plot  it  on 
the  diagram,  and  read  at  once  the  side  cuts,  g  k,  en. 


58  RAILWAY   CURVES   AND    LOCATION. 

the  area  is  equal  to  that  of  the  trapezoid  e  n  k  g,  minus 
the  two  triangles  e  n  h,  mk  g. 


Calling  e  ft,  or  its  equal  n  li,  a,  m  k  or  g  k,  5,  and  half 
the  breadth  of  the  roadway,  c,  we  have  the  triangle  mk  g 

=  b  X  77  =  «-•    The  triangle  e  n  h  is  in  like  manner  =  —, 

22  2 

and  the  area  of  the  trapezoid  =  a-\-b-{-2cX  — J— . 

2 

Then  the  area  required  =  a-j-5-|-2tfX  —- ~ — 


2 
2  ab  -f  2  ac  -f  2bc 


=  (a  -f  b)  c  +  a.5. 


Should  the  slopes  be  J  to  1,  as  in  rock  cut,  the  area 
might,  in  similar-wise,  be  shown  equal  to  (a  -f-  b)  c  -\ — ^-, 
and  that  of  embankment,  where  the  slopes  are  1J  to  1, 
equal  to  (a  -f-  #)  c  -j ^-.  Wherefore,  we  arrive  at  the 

following  general  rule  for  finding  cross-sectional  areas, 
where  the  ground  surface  is  a  regular  declivity : 

Multiply  the  sum  of  the  side  cuttings  by  half  the  breadth 
of  the  roadway,  and  mark  the  product.  Multiply  the  pro- 
duct of  the  side  cuttings  by  the  ratio  of  the  side  slopes  to 
1,  and  add  this  result  to  the  previous  product  marked  for 
the  area. 

Example  1. — Earth  excavation — road  bed  32  feet,  side 


OF  EXCAVATION   AND   EMBANKMENT.  59 

slopes  1  to  1,  right  cutting  12  feet,  left  cutting  3  feet. 
Required  the  cross-sectional  area.  Here  (a  -f-  b)  c  -\-  a  b 
=  (12  +  3)  16  -f  12  X  3  =  240  +  36  =  276  square 
feet,  the  area  required. 

Example  2.  —  Rock  cut  —  road  bed  28  feet,  side 
slopes  J  to  1,  cuttings  as  before.  Required  the  area. 

Here  (a  -f  b)  c  +  t-*J.  =  210    +^  =   219   square 

feet,  the  area. 

Example  3.  —  Embankment  —  roadway  27  feet,  side 
slopes  1|  to  1,  cuttings  as  before.  Here  («  +  #)<?  + 

3  ^2X  ^  =  202-5.  +  54  =  256-5  square  feet,  the  cross- 

sectional  area. 

Other  formulae  might  be  given  for  varying  surface  slopes, 
but  they  would  involve  a  simple  matter,  and  require  more 
time  for  calculation  than  a  division  of  the  diagram  area 
into  triangles. 

To  find  the  cubic  content  of  an  excavation  or  embank- 
ment :  — 

Multiply  half  the  sum  of  the  two  end  areas  by  the 
distance  between  them  ;  thus  :  supposing  282  and  310 
to  be  the  end  areas,  and  100  feet  the  distance,  we  have 
(282  +  310) 


yards.  Or  we  may  multiply  the  sum  of  the  end  areas  by 
100,  and  divide  the  product  by  6  and  9,  the  factors  of  54, 
which  is  the  number  of  cubic  feet  contained  in  two  cubic 

yards  ;  thus,  5?|??  =  9866-6,  and  this  divided  by  9  = 
1096-3,  as  above. 


60 


RAILWAY    CURVES    AND    LOCATION. 


PRISMOIDAL   FORMULA. 

To  the  two  end  areas  add  four  times  the  middle  area, 
and  multiply  the  sum  by  one-sixth  of  the  length  of  the  pris- 
moid.  Thus  :  from  the  foregoing  example,  the  sum  of  the 
end  areas,  592,  added  to  four  times  their  mean,  1184, 
gives  1776,  and  1776  X  16-7  =  29659-2  cubic  feet,  = 
1098'5  cubic  yards.  The  former  rule  is  approximate — 
sufficiently  so  for  rough  preliminary  calculations,  but  where 
strict  correctness  is  required,  as  in  a  final  field  estimate, 
the  prismoidal  formula  should  be  usefl.  It  applies  to  all 
solid  bodies  with  plane  faces  and  parallel  ends. 


ARTICLE  XXIV. 


SIDE    STAKING. 


THE  object  in  side  staking  is  to  find  the  point  where 
the  surface  of  the  ground  intersects  the  slope  of  the  road 


formation.  For  performing  this  work  with  level  and  rod, 
place  the  instrument  in  such  a  position  as  to  command  as 
many  stations  as  possible,  whether  they  be  on  the  upper 


SIDE   STAKING  61 

or  lower  side,  and  ascertain  by  transfers  from  the  bench 
the  height  of  your  instrument  with  reference  to  grade. 
This  will  facilitate  operations  by  giving  usually  small  num- 
bers to  work  with.  Knowing  your  rate  of  grade,  subtract 
or  add,  as  the  case  may  be,  as  you  change  from  station  to 
station. 

Having  the  level  in  place,  we  will  suppose  on  an  exca- 
vation, our  object  is  first  to  place  the  lower  side  stake. 
Let  the  width  of  the  road  bed  be  26  feet,  and  the  side 
slopes  1  to  1.  Suppose  the  instrument  to  be  11  feet  above 
grade  and  the  centre  cut  5  feet,  which  latter  is  already 
recorded  in  the  field  book  for  construction  duties.  Look- 
ing at  the  fall  of  the  hill  we  judge  that  at  the  distance  of 
17  feet  from  the  centre  stake  the  descent  is  1'5  feet,  which 
would  give  us  3-5  feet  cutting.  We  take  an  observation 
at  that  point  and  find  that  the  rod  reads  10  feet,  which, 
subtracted  from  our  height  above  grade,  leaves  us  only  one 
foot  cutting,  and  shows  that  our  judgment  has  been  at 
fault.  The  distance  measured  is  too  much,  for  were  we  to 
increase  the  distance  we  would  reduce  the  cutting,  and  it 
is  therefore  evident  that  we  are  too  far  from  the  centre 
Let  us  then  make  the  distance  from  the  centre  stake  15 
feet,  and  take  another  sight,  when  the  rod  reads  9,  show- 
ing 2  feet  cutting,  which,  added  to  half  the  road  formation, 
slope  being  1  to  1,  equals  the  distance  measured.  The 
stake  is  therefore  correct.  The  same  process  applies  to 
the  upper  side. 

The  only  difference  in  staking  out  for  banks  is  that  you 
add  1J  times  the  height  of  the  bank  to  the  distance  mea- 
sured, or,  as  a  general  rule,  the  height  of  the  bank  multi- 
plied by  the  ratio  of  the  side  slopes  to  unity. 

Some  judgment  is  required  to  be  expeditious  in  this 
work,  which  is  obtained  only  by  experience. 

6 


62 


RAILWAY   CURVES   AND   LOCATION. 


The  following  is  a  common  form  of  field  record  for  con 
struction : 


,«»* 

jji8t. 

Left  Side. 

Centre. 

•      i 
Right  Sido 

Course. 

Mag. 
Course. 

Elevation 

Grade. 

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A 

B 

Dist. 

AorB 

S-ll  A 

530 

100 

N20°OOW 

N  20-05  W 

+  560-4 

+  555-4 

15-0 

2-0  A 

5-0 

21-0 

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767754 

03723354-561300 

1145-9153 

3 

58 

374 

999 

2 

B8 

871068  28-166422  55'441517 

1718-8732 

2 

59 

426 

I-  0000  000 

1 

5! 

975523   399397  56'350590 

3437-7467 

1 

60 

4771     000 

0 

e< 

19-091137   63625357-281962 

Infinite. 

0 

' 

IQ     0° 

' 

/ 

3°   1   2°     1° 

0° 

NAT.  COSINK.                       NAT.  COT  AN. 

89 

TABLE  OF  RADII,  &c.~-  Chord  100  Feet. 

An<j;l<>  of 

Radius 

Def.  dis 

Angle  o 

Radius 

Def.  dis 

Angle  ol 

'    Radius    Def.  dist 

LL'tk-.vn 

in  feet. 

in  feet 

Detiex'n 

in  feet. 

in  feet 

Deflex'n 

in  feet. 

in  feet. 

O         / 

0         / 

0        / 

5  |68760-0     -145 

3  25 

1677-0 

5-962 

12  U 

468-7 

21.360 

,       10 

34380-0    -291 

30 

1637-0 

6-108 

3C 

459-3 

21-790  ! 

15 

22920-0    -436 

35 

1599-0 

6-253 

o    45 

450-3  22-210 

20  117190-01    -581 

40 

1563-0 

6-398 

13 

441-7  22-640 

25    13752-0]    -727 

45 

1528-0 

6-544 

15 

433-4  23-070 

30 
35 

11460-01    -872 
9823-0  1-017 

50 
o  55 

1495-0 
1463-0 

6-689 
6-835 

3C 

o    45 

425-5  23-510 
417-7  i23-940  i 

40 

8595-0  1-163 

4  . 

1433.0!  6-980 

14 

410-3  24-370  , 

45 

7640-0  1-308 

15 

1348-0   7-416 

15 

403-1  J24-810  i 

!      50 

6876-0  1-453 

30 

1274-0   7-853 

30 

396-2  ;25-24</  ' 

i  o  55 

6251-0  1-600 

o  45 

1207-0   8-289 

o    45 

389-6  ;25-670 

1 

5730-0  1-745 

5 

1146-0 

8-722 

15 

383-  U26-1  10 

5 

5289-01  1-890 

15 

1092-0 

9-159 

15 

376-9  126-520 

10 

4912-0 

2-036 

30 

1042-0 

9-595 

30 

370-8  J26-940 

15 

4584-0 

2-181 

o  45 

996-8 

10-030 

o    45 

365-0  27-370 

20 

4298-0 

2-327 

6 

955-4 

10-470 

16 

359-3  27-830 

25 

4045-0  2-472 

15 

917-0 

10-900 

o    30 

348-4  28-700 

30 
35 

3820-0  2-618 
3619-0  2-763 

30 
o  45 

882-0 
849-3 

11-340 
11-780 

17 

o    30 

338-3 

328-7 

29-560 
30-430  i 

!      40 

3438-0 

2-908 

7 

819-0 

12-210 

18 

319-6 

31-290 

45 

3274-0 

3-054 

15 

790-8 

12-640 

o    30 

311-0 

32-500 

50 

3125-0 

3-199 

30 

764-5 

13-080 

19 

302-9 

33010 

o  55 

2990-0 

3-345 

o  45 

739-9 

13-510 

o    30 

295-3 

33-870 

2 

2865-0 

3-490 

8 

716-8 

13-950 

20 

287-9 

34-730 

5 

2750-0 

3-635 

15 

695-1 

14-380 

21 

274.4 

36-440 

10 

2644-0 

3-781 

30 

674-6 

14-810 

22 

262-0 

38-150 

15 

2547-0 

3-926 

o  45 

655-5 

15-250 

23 

250-8 

39-870 

20 

2456-0 

4-072 

9 

637-3 

15-680 

24 

240-5 

41-580 

25 

2371-0 

4-217 

15 

620-2 

16-120 

25 

231-0 

43-280 

30 

2292-0 

4-363 

30 

603-8 

16-550 

26 

222-3 

44-980 

35 

2218-0 

4-508 

o    45 

588-4 

16-990 

27 

214-2 

46-680 

40 

21490 

4-653 

10 

573-7J17-430 

28 

206-7 

48-380  ! 

45 

2084-0 

4-799 

15 

559-7  17-870 

29 

199-7 

50-070 

50 

2023-0!  4-9  14 

30 

546.4 

18-300 

30 

193-2 

51-760 

!  o  55 

1965-0  5-090 

o    45 

533-8J18-730 

31 

187-1 

53-450 

3 

1910-0 

5-235 

11 

521-719-170 

32 

181-4 

55-130 

5 

1859-0  5-380 

15 

510-119-610 

33 

176-0 

56-800 

10 

1810-0  5-526 

30 

499-120-050 

34 

171-0 

58/470 

15 

1763-0!  5-671 

o    45 

488-520-500 

35 

166-3 

60-140" 

20 

1719-0 

5-817 

12 

478-3j20-940 

36 

.161-8 

61-800  j 

8* 

90 

TABLE  OF  LONG  CHORDS. 

Raclics  in  feet. 

Angle  of 
Deflection. 

Length 
1  Station. 

of  Chord  in  f< 
2  Stations. 

jet  required  tc 
3  Stations. 

subtend 
4  Stations. 

5730-0 

1° 

100 

200-0 

300-0 

400-0 

4584-0 

* 

100 

200-0 

300-0 

399-9 

3820-0 

| 

100 

200-0 

300-0 

399-9 

3274-0 

100 

200-0 

300-0 

399-8 

2865-0 

2°f 

100 

200-0 

299-9 

399-7 

I      2547-0 

* 

100 

200-0 

299-9 

399-6 

2292-0 

100 

200-0 

299-8 

399-5 

2084-0 

f. 

100 

200-0 

299-8 

399-4 

1910-0 

3° 

100 

200-0 

299-7 

399-3 

1763-0 

i 

100 

200-0 

299-7 

399-2 

.       1637-0 

| 

100 

200-0 

299-6 

399-1 

1528-0 

100 

200-0 

299-6 

399-0 

1433-0 

4°  f 

100 

199-9 

299-6 

398-9 

1348-0 

\ 

100 

199-9 

299-5 

398-7 

1274-0 

100 

199-9 

299-4 

398-5 

1207-0 

I 

100 

199-9 

299-3 

398-3 

1146-0 

5° 

100 

199-9 

299-2 

398-0 

1092-0 

k 

100 

199-8 

299-1 

397-8 

1042-0 

100 

199-8 

299-0 

397-6 

1        996-8 

! 

100 

199-7 

298-9 

397-5 

955.4 

6° 

100 

199-7 

298-8 

397-3 

917-0 

i 

100 

199-7 

298-7 

397-0 

882-0 

£ 

100 

199-7 

298-6 

396-7 

849-3 

1 

100 

19^fc 

298-5 

396-5 

819-0 

7° 

100 

199-6 

298-4 

396-2 

790-8 

i 

100 

199-6 

298-3 

396-0 

764-5 

* 

100 

199-6 

298-2 

395-7 

739-9 

1 

100 

199-6 

298-1 

395-4 

1        716-8 

8° 

100 

199-6 

298-0 

395-1 

695-1 

i 

100 

199-5 

297-9 

394-8 

674-6 

100 

199-5 

297-8 

394-5 

655-5 

3. 

100 

199-4 

297-7 

394-3 

637-3 

9° 

100 

199-4 

297-5 

394-1 

620-2 

i 

100 

199-4 

297-4 

393-7 

603-8 

100 

199-3 

297-3 

393-2 

588-4 

1 

100 

199-2 

297-2 

392-8 

573-7 

10° 

100 

199-2 

297-0 

392-4 

: 

TABLE  OF  ORDINATES. 

Ordinates  10  feet  apart.—  Chord  100  feet. 

Distances  of  the  Ordinates  from  the  end  of  the  100  feet  Chord. 

Deflexion  Angle 
in  Degrees  and 

50  feet. 

40  feet. 

30  feet. 

20  feet. 

10  feet. 

Lengths  of  Ordinates  in  feet. 

o       / 

5 

•018 

•017 

•015 

•012 

•006 

10 

•036 

•035 

•031 

•023       '      -013 

15 

•054 

•052 

•046 

•035 

•019 

20 

•073 

•070 

•061 

•047 

•026 

25 

•091 

•087 

•076 

•058 

•032 

30 

•109 

•105 

•092 

•070 

•039       i 

35 

•127 

•123 

•108 

•082 

•045 

40 

•145 

•140 

•123 

•093 

•052       i 

45 

•163 

•157 

•137 

•105 

•058 

50 

•182 

•175 

•153 

•117 

•065 

o     55 

•200 

•192 

•168 

•128 

•071 

1 

•218 

•209 

•183 

•140 

•078 

5 

•236 

•226 

•198 

•152 

•085 

10 

•254 

•244 

•214 

•163 

•091 

15 

•273 

•261 

•229 

•175 

•098 

20 

•291 

•279 

•244 

•187 

•104      i 

25 

•309 

•296 

•259 

•198 

•111 

30 

•327 

•314 

•275 

•210 

•117 

35 

•345 

•331 

•290 

*221 

•124 

40 

•364 

•349 

•305 

•233 

•130 

45 

•382 

•366 

•321 

•245 

•137 

50 

•400 

'-384 

•336 

•256 

•144 

o     55 

•418 

•401 

•351 

•268 

•150 

2 

•436 

•419 

•366 

•280 

•157 

5 

•454 

•436 

•382 

•291 

•163 

10 

•473 

•454 

•397 

•303    .        -170 

15 

•491 

•471 

•412 

•315            -17' 

20 

•509 

•489 

•428 

•326              1. 

25 

•527 

•506 

•443 

•338 

•190 

30 

•545 

•524 

•458 

•350 

•196 

35 

•564 

•541 

•474 

•361 

•203 

40 

•582 

•559 

•489 

•373 

•209 

45 

•600 

•576 

•504 

•384 

•216       ; 

.50 

•618 

•594 

•519 

•396 

•222 

o     55 

•636 

•611 

•535 

•408 

•229       i 

3 

•654 

•629 

•550 

•419 

•235 

5 

•673 

•646 

•565 

•431 

•242 

10 

•691 

•664 

•581 

•443 

•249 

15 

•709 

•681 

•596 

•454 

•255 

!              20 

•727 

•699 

•611 

•466 

•262 

;              25 

j  - 

•745 

•716 

•627 

•478 

•268 

1                                                                                                                                                1 

92 

TABLE  OF  ORDINATES.—  CONTINUED. 

Ordinates  10  feet  apart.—  Chord  100  feet. 

Distances  of  the  Ordinates  from  the  end  of  the  100  feet  Chord. 

Deflexion  Angle 
in  Degrees  and 
Minutes. 

60  feet. 

40  feet. 

30  feet. 

20  feet. 

10  feet 

Lengths  of  Ordinates  in  feet. 

o         / 

3    30 

•764 

•734 

•642 

•489 

•275 

35 

•782 

•751 

•657 

•501 

•281 

!              40 

•800 

•769 

•673 

•512 

•288 

45 

•818 

•786 

•688 

•524 

•294 

50 

•836 

•804 

•703 

•536 

•301 

o     55 

•854 

•821 

•718 

•547 

•308 

4 

•873 

•839 

•734 

•559 

•314 

15 

•927 

•891 

•780 

•594 

•334 

30 

•981 

•944 

•825 

•629 

•354 

o     45 

1-036 

•996 

•871 

•664 

•373 

5 

1-091 

1-048 

•917 

•699 

•393 

15 

1-146 

1-100 

•963 

•734 

•413 

30 

1-200 

1-153 

1-009 

•769 

•432 

o     45 

1-255 

1-205 

1-055 

•804 

•452 

6 

1-309 

1-258 

1-100 

•839 

•472 

15 

1-364 

1-310 

1-146 

•874 

•492 

30 

1-419 

1-362 

1-192 

•909 

•511 

o     45 

1-473 

1-415 

1-238 

•944 

•531 

7 

1-528 

1-467 

1-284 

•979 

•551 

15 

1-582 

1-520 

1-330 

1-014 

•570 

30 

1-637 

1-572 

1-375 

1-048 

•590 

o     45 

1-692 

1-624 

1-421 

1-083  . 

•610 

8 

1-746 

1-677 

1-467 

1-118 

•629 

15 

1-801 

1-729 

1-513 

1-153 

•649 

30 

1-855 

1-782 

1-559 

1-188 

•609 

o     45   - 

1-910 

1-834 

1-605 

1-223 

•689 

9 

1-965 

1-886 

1-651 

1-258 

•708 

15 

2-019 

1-939 

1-696 

1-293 

•728 

30 

2-074 

1-991 

1-742 

1-328 

•748 

I        o      45 

2-128 

2-044 

1-788 

1-363 

•767 

10 

2-183 

2-096 

1-834 

1-398 

•787      [ 

15 

2-238 

2-148 

1-880 

1-433 

•807 

30 

2-292 

2-201 

1-926 

1-468 

•827 

o      45 

2-347 

2-254 

1-972 

1-503 

•846 

11 

2-401 

2-306 

2-018 

1-538 

•866 

15 

2-456 

2-359 

2-064 

1-574 

•886 

30 

2-511 

2-411 

2-110 

1-609 

•906 

o      45 

2-566 

2-464 

2-156 

1-644 

•926 

12 

2-620 

2-516 

2-203 

1-680 

•946 

15 

2-675 

2-569 

2-249 

1-715 

•966 

30 

2-730 

2-621 

2.295 

1-750 

•985 

if 

93 

TABLE  OF  ORDINATES.—  CONTINUED. 

Ordinates  10  feet  apart.—  Chord  100  feet. 

Distances  of  the  Ordinates  from  the  end  of  the  100  feet  Chord.       ; 

Deflexion  Angle 
iu  Degrees  and 

50  feet. 

40  feet. 

30  feet. 

20  feet. 

10  feet. 

Minutes. 

Lengths  of  Ordinates  in  feet. 

o         / 

12    45 

2785 

2-674 

2-341 

1-785 

1-005 

13 

2-839 

2-726 

2-387 

1-820 

1-025     1 

15 

2-894 

2-779 

2-433 

1-855 

1-045 

30 

2-949 

2-832 

2-479 

1-891 

1-065 

o     45 

3-000 

2-884 

2-525 

1-926 

1-085 

14 

3-058 

2-937 

2-571 

1-961 

1-105 

15 

3-113 

2-989 

2-618 

1-996 

1-124 

30 

3-168 

3-042 

2-664 

2-031 

1-144 

o     45 

3-222 

3-094 

2-710 

2-067 

1  164 

15 

3-277 

3-147 

2-756 

2-102 

1-184 

15 

3-332 

3-200 

2-802 

2-137 

1-204 

30 

3-387 

3-252 

2-848 

2-172 

1-224 

o     45 

3-442 

3-305 

2-895 

2-208 

1-244 

16 

3-496 

3-358 

2-941 

2-243 

1-264 

o     30 

3-606 

3-463 

3-033 

2-314 

1-304 

17 

3-716 

3-569 

3-125 

2-384 

1-344 

o      30 

3-826 

3-674 

3-218 

2-455 

1-384 

i      18 

3-935 

3-779 

3-310 

2-525 

1-424 

i       o     30 

4-045 

3-885 

3-403 

2-596 

1-464 

19 

4-155 

3-990 

3-495 

2-666 

1-504 

o'   30 

4-265 

4-096 

3-588 

2-737          1-544 

20 

4-375 

4-201 

3-680 

2-808 

1-583 

94 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

1 

1 

1-0000 

44 

1936 

6-6332 

2 

4 

1-4142 

45 

2025 

6-7082 

i   3 

9 

1-7320 

46 

2116 

6-7823 

4 

16 

2-0000 

47 

2209 

6-8556 

5 

25 

2-2360 

48 

2304 

6-9282   ' 

i   6 

36 

2-4495 

49 

2401 

7-0000 

7 

49 

2-6457 

50 

2500 

7-0711 

8 

64 

2-8284 

51 

2601 

7-1414 

9 

81 

3-0000 

52 

2704 

7-2111 

10 

100 

3-1623 

53 

2809 

7-2801 

11 

121 

3-3166 

54 

2916 

7-3485 

12 

144 

3-4641 

55 

3025 

7-4162 

13 

169 

3-6055 

56 

3136 

7-4833 

14 

196 

3*7416 

57 

3249 

7-5498 

15 

225 

3-8730 

58 

3364 

7-6158 

16 

256 

4-0000 

59 

3481 

7-6811 

17 

289 

4-1231 

60 

3600 

7-7460 

18 

324 

4-2426 

61 

3721 

7-8102 

19 

3G1 

4-3589 

62 

3844 

7-8740 

20 

400 

4-4721 

63 

3969 

7-9372 

21 

441 

4-5826 

64 

4096 

8-0000 

22 

484 

4-6904 

65 

4225 

8-0628 

,  23 

529 

4-7958 

66 

4356 

8-1240 

24 

576 

4-8990 

67 

4489 

8-1853 

25 

625 

5-0000 

68 

4624 

8-2462 

26 

676 

5-0990 

69 

4761 

8-3066 

27 

729 

5-196L 

70 

4900 

8-3666 

28 

784 

5-2915 

71 

5041 

8-4261 

29 

841 

5-3852 

72 

5184 

8-4853 

30 

900 

5-4772 

73 

5329 

8-5440 

31 

961 

5-5678 

74 

5476 

8-6023 

32 

1024 

5-6568 

75 

5625 

8-6603 

33 

1089 

5-7446 

76 

5776 

8-7178 

34 

1156 

5-8309 

77 

5929 

8-7750 

35 

1225 

5-9161 

78 

6084 

8-8318 

36 

1296 

6-0000 

79 

6241 

8-8882 

37 

1369 

6-0828 

80 

6400 

8-9443   ; 

38 

1444 

6-1644 

81 

6561 

9-0000 

39 

1521 

6-2450 

82 

6724 

9-0554   i 

40 

1600 

6-3246 

83 

6889 

9-1104 

41 

1681 

6-4031 

84 

7056 

9-1651 

42 

1764 

6-4807 

85 

7225 

9-2195 

43 

1849 

6-5574 

86 

7396 

9-2736   i 

i 

95 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

!   87 

7569 

9-3274 

130 

16900 

11-4017 

88 

7744 

9-3808 

131 

17161 

11-4455 

89 

7921 

9-4340 

132 

17424 

11-4891 

90 

8100 

9-4868 

133 

17689 

11-5326 

91 

8281 

9-5394 

134 

17956 

11-5758 

92 

8464 

9-5917 

135 

18225 

11-6189 

93 

8649 

9-6436 

136 

18496 

11-6619 

94 

8836 

9-6954 

137 

18769 

11-7047 

95 

9025 

9-7468 

138 

19044 

11-7473 

96  i    9216 

9-7979 

139 

19321 

11-7898 

97 

9409 

9-8488 

140 

19600 

11-8322 

98 

9604 

9-8995 

141 

19881 

11-8743 

99 

9801 

9-9499 

142 

20164 

11-9164 

100 

10000 

10-0000 

143 

20449 

11-9583 

101 

10201 

10-0499 

144 

20736 

12-0000 

102 

10404 

10-0995 

145 

21025 

12-0416 

103 

10609 

10-1489 

146 

21316 

12-0830 

104 

10816 

10-1980 

147 

21609 

12-1244 

105 

11025 

10-2469 

148 

21904 

12-1655 

106 

11236 

10-2956 

149 

22201 

12-2065 

107 

11449 

10-3440 

150 

22500 

12-2474 

108 

11664 

10-3923 

151 

22801 

12-2882 

109 

11881 

10-4403 

152 

23104 

12-3288 

110 

12100 

10-4881 

153 

23409 

12-3693 

111 

12321 

10-5356 

154 

23716 

12-4097 

112 

12544 

10-5830 

155 

24025 

12-4499 

113 

12769 

10-6301 

156 

24336 

12-4900 

114 

12996 

10-6771 

157 

24649 

12-5300 

115 

13225 

10-7238 

158 

24964 

12-5700 

116 

13456 

10-7703 

159 

25281 

12-6095 

117 

13689 

10-8166 

160 

25600 

12-6491 

118 

13924 

10-8628 

161 

25921 

12-6886   ! 

119 

14161 

10-9087 

162 

26244 

12-7279   , 

120 

14400 

10-9544 

163 

26569 

12-7671 

i   121 

14641 

11-0000 

164 

26896 

12-8062   | 

122 

14884 

11-0454 

165 

27225 

12-8452 

123 

15129 

11-0905 

166 

27556 

12-8841 

124 

15376 

11-1355 

167 

•  27889 

12-9228 

125 

15625 

11-1803 

168 

28224 

129615 

126 

15876 

11.2250 

169 

28561 

13-0000 

127 

16129 

11-2694 

170 

28900 

13-0384 

128 

16384 

11-3137 

171 

29241 

13-0767 

129 

16641 

11-3578 

172 

29584 

13-1149   1 

96 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Koots. 

173 

29929 

13-1529 

216 

46656 

14-6969 

174 

30276 

13-1909 

217 

47089 

14-7309 

175 

30625 

13-2287 

218 

47524 

14-7648 

I   176 

30976 

13-2665 

219 

47961 

14-7986 

177 

31329 

13-3041 

220 

48400 

14-8324 

178 

31684 

13-3417 

221 

48841 

14-8661 

179 

32041 

13-3791 

222 

49284 

14-8997 

180 

32400 

13-4164 

223 

49729 

14-9332 

181 

32761 

13-4536 

224 

50176 

14-9666 

182 

33124 

13-4907 

225 

50625 

15-0000 

183 

33489 

13-5277 

226 

51076 

15-0333 

184 

33856 

13-5647 

227 

51529 

15-0665 

185 

34225 

13-6015 

228 

51984 

15-0997 

186 

34596 

13-6382 

229 

52441 

15-1327 

187 

34969 

13-6748 

230 

52900 

15-1657 

188 

35344 

13-7113 

231 

53361 

15-1987 

189 

35721 

13-7477 

232 

53824 

15-2315 

190 

36100 

13-7840 

233 

54289 

15-2643 

191 

36481 

13-8203 

234 

54756 

15-2970 

192 

36864 

13-8564 

235 

55225 

15-3297 

193 

37249 

13-8924 

236 

55696 

15-3623 

194 

37636 

13-9284 

237 

56169 

15-3948 

195 

38025 

13-9642 

238 

56644 

15-4272 

196 

38416 

14-0000 

239 

57121 

15-4596 

197 

38809 

14-0357 

240 

57600 

15-4919 

198 

39204 

14-0712 

241 

58081 

15-5242 

199 

39601 

14-1067 

242 

58564 

15-5563 

200 

40000 

14-1421 

243 

59049 

15-5885 

201 

40401 

14-1774 

244 

59536 

15-6205 

202 

40804 

14-2127 

245 

60025 

15-6525 

203 

41209 

14-2478 

246 

60516 

15-6844 

204 

41616 

14-2828 

247 

61009 

15-7162 

205 

42025 

14-3178 

248 

61504 

15-7480 

206 

42436 

14-3527 

249 

62001 

15-7797 

207 

42849 

14-3874 

250 

62500 

15-8114 

208 

43264 

14-4222 

251 

63001 

15-8430 

209 

43681 

14-4568 

252 

63504 

15-8745 

210 

44100 

14-4914 

253 

64009 

15-9060 

211 

44521 

14-5258 

254 

64516 

15-9374 

212 

44944 

14-5602 

255 

65025 

15-9687 

213 

45369 

14-5945 

256 

65536 

16-0000 

214 

45796 

14-6287 

257 

66049 

16-0312 

215 

46225 

14-6629 

258 

66564 

16-0624 

97 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

259 

67081 

16.0935 

302 

91204 

17-3781 

260 

67600 

16-1245 

303 

91809 

17-4069 

261 

68121 

16-1555 

304 

92416 

17-4356 

262 

68644 

16-1864 

305 

93025 

17-4642 

263 

69169 

16-2173 

306 

93636 

17-4928 

264 

69696 

16-2481 

307 

94249 

17-5214 

265 

70225 

16-2788 

308 

94864 

17-5499 

266 

70756 

16-3095 

309 

95481 

17-5784 

267 

71289 

16-3401 

310 

96100 

17-6068 

268 

71824  ' 

16-3707 

311 

96721 

17-6352 

269 

72361 

16-4012 

312 

97344 

17-6635 

270 

72900 

16-4317 

313 

97969 

17-6918 

271 

73441 

16-4621 

314 

98596 

17-7200 

272 

73984 

16-4924 

315 

99225 

17-7482 

273 

74529 

16-5227 

316 

99856 

17-7764 

274 

75076 

16-5529 

317 

100489 

17-S045 

275 

75625 

16-5831 

318 

101124 

17-8325 

276 

76176 

16-6132 

319 

101761 

17-8606 

277 

76729 

16-6433 

320 

102400 

17-8885 

278 

77284 

16-6733 

321 

103041 

17-9165 

279 

77841 

16-7033 

322 

103684 

17-9444 

280 

78400 

16-7332 

323 

104329 

17-9722 

281 

78961 

16-7630 

324 

104976 

18*0000 

282 

79524 

16-7928 

325 

105625 

18-0277 

283 

80089 

16-8226 

326 

106276 

18-0555 

284 

80656 

16-8523 

327 

106929 

18-0831 

285 

81225 

16-8819 

328 

107584 

18-1108 

286 

81796 

16-9115 

329 

108241 

18-1384 

287 

82369 

16-9411 

330 

108900 

18-1659 

288 

82944 

16-9706 

331 

109561 

18-1934 

289 

83521 

17-0000 

332 

110224 

18-2209 

290 

84100 

17-0294 

333 

110889 

18-2483 

291 

84681 

17-0587 

334 

111556 

18-2757 

292 

85264 

17-0880 

aas 

112225 

18-3030 

293 

85849 

17-1172 

336 

112896 

18-3303 

294 

86436 

17-1464 

337 

113569 

18-3576 

295 

87025 

17-1756 

338 

114244 

18-3848 

296 

87616 

17*2046 

339 

114921 

18-4119 

297 

88209 

17-2337 

340 

115600 

18-4391 

298 

88804 

17-2627 

341 

116281 

18-4662 

299 

89401 

17-2916 

342 

116964 

18-4932 

300 

90000 

17-3205 

343 

117649 

18-5203 

301 

90601 

17-3493 

344 

118336 

18-5472 

98 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

345 

119025 

18-5742 

388 

150544 

19-6977 

346 

119716 

18-6011 

389 

151321 

19-7231 

347 

120409 

18-6279 

390 

152100 

19-7484 

348 

121104 

18-6548 

391 

152881 

19-7737 

349 

121801 

18-6815 

392 

153664 

19-7990 

350 

122500 

18-7083 

393 

154449 

19-8242 

351 

123201 

18-7350 

394 

155236 

19-8494 

352 

123904 

18-7617 

395 

156025 

19-8746 

353 

124609 

18-7883 

396 

156816 

19-8997 

354 

125316 

18-8149 

397 

157609 

19-9248 

355 

126025 

18-8414 

398 

158404 

19-9499 

356 

126736 

18-8680 

399 

159201 

19-9750 

357 

127449 

18-8944 

400 

160000 

20-0000 

358 

128164 

18-9209 

401 

160801 

20-0250 

359 

128881 

18-9473 

402 

161604 

20-0499 

360 

129600 

18-9737 

403 

162409 

20-0749 

361 

130321 

19-0000 

404 

163216 

20-0997 

362 

131044 

19-0263 

405 

164025 

20-1246 

363 

131769 

19-0526 

406 

164836 

20-1494 

364 

132496 

19-0788 

407 

165649 

20-1742 

365 

133225 

19-1050 

408 

166464 

EO-1990 

366 

133956 

19-1311 

409 

167281 

20-2237  - 

367 

134689 

19-1572 

410 

168100 

20-2485 

368 

135424 

19-1833 

411 

168921 

20-2731 

369 

136161 

19-2094 

412 

169744 

20-2978 

370 

136900 

19-2354 

413 

170569 

20-3224 

371 

137641 

19-2614 

414 

171396 

20-3470 

372 

138384 

19-2873 

415 

172225 

20-3715 

373 

139129 

19-3132 

416 

173056 

20-3961 

374 

139876 

19-3391 

417 

173889 

20-4206 

375 

140625 

19-3649 

418 

174724 

20-4450 

376 

141376 

19-3907 

419 

175561 

20-4695 

377 

142129 

19-4165 

420 

176400 

20-4939 

378 

142884 

19-4422 

421 

177241 

20-5183 

379 

143641 

19-4679 

422 

178084 

20-5426 

380 

144400 

19-4936 

423 

178929 

20-5670 

381 

145161 

19-5192 

424 

179776 

20-5913 

382 

145924 

19-5448 

425 

180625 

20-6155 

383 

146689 

19-5704 

426 

181476 

20-6398 

384 

147456 

19-5959 

427 

182329 

20-6640 

385 

148225 

19-6214 

428 

183184 

20-6882 

386 

148996 

19-6469 

429 

184041 

20-7123 

387 

149769 

19-6723 

430 

184900 

20-7364 

11  

99 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

431 

185761 

20-7605 

474 

224676 

21-7715 

432 

186624 

20-7846 

475 

225625 

21-7945 

433 

187489 

20-8086 

476 

226576 

21-8174 

434 

188356 

20-8327 

477 

227529 

21-8403 

435 

189225 

20-8566 

478 

228484 

21-8632  • 

43G 

190096 

20-8806 

479 

229441 

21-8861 

437 

190969 

20-9045 

480 

230400 

21-9089 

438 

191844 

20-9284 

481 

231361 

21-9317 

439 

192721 

20-9523 

482 

232324 

21-9545   I 

440 

193600 

20-9762 

483 

233289 

21-9773   1 

441 

194481 

21-0000 

484 

234256 

22-0000 

442 

195364 

21-0238 

485 

235225 

22-0227 

443 

196249 

21-0476 

486 

236196 

22-0454 

444 

197136 

21-0713 

487 

237169 

22-0689 

445 

198025 

21-0950 

488 

238144 

22-0907 

446 

198916 

21-1187 

489 

239121 

22-1133 

447 

199809 

21-1424 

490 

240100 

22-1359 

448 

200704 

21-1660 

491 

241081 

22-1585 

449 

201601 

21-1896 

492 

242064 

22-1811 

450 

202500 

21-2132 

493 

243049 

22-2036 

451 

203401 

21-2368 

494 

244036 

22-2261 

452 

204304 

21-2603 

495 

245025 

22-2486   j 

453 

205209 

21-2838 

496 

246016 

22-2711 

454 

206116 

21-3073 

497 

247009 

22-2935 

455 

207025 

21-3307 

498 

248004 

22-3159 

456 

207936 

21-3542 

499 

249001 

22-3383 

457 

208849 

21-3776 

500 

250000 

22-3607 

458 

209764 

21-4009 

501 

251001 

22-3830 

459 

210681 

21-4243 

502 

252004 

22-4054 

460 

211600 

21-4476 

503 

253009 

22-4277 

461 

212521 

21-4709 

504 

254016 

22-4499 

462 

213444 

21-4942 

505 

255025 

22-4722 

463 

214369 

21-5174 

506 

256036 

22-4944   i 

464 

215296 

21-5407 

507 

257049 

22-5167 

465 

216225 

21-5639 

508 

258064 

22-5388   1 

466 

217156 

21-5870 

509 

259081 

22-5610 

467 

218089 

21-6102 

510 

260100 

22-5832   I 

1  468 

219024 

21-6333 

511 

261121 

22-6053 

469 

219961 

21-6564 

512 

262144 

22-6274 

470 

220900 

21-6795 

513 

263169 

22-6495 

471 

221841 

21-7025 

514 

264196 

22-6716   | 

472 

222784 

21-7256 

515 

265225 

22-6936 

473 

223729 

21-7486 

516 

266256 

22-7156 

100 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots.  ! 

i  517 

267289 

22-7376 

560 

313600 

23-6643 

1  518 

268324 

22-7596 

561 

314721 

23-6854 

519 

269361 

22-7816 

562 

315844 

23-7065 

520 

270400 

22-8035 

563 

316969 

23-7276 

•521 

271441 

22-8254 

564 

318096 

23-7487 

522 

272484 

22-8473 

565 

319225 

23-7697 

523 

273529 

22-8692 

566 

320356 

23-7907 

524 

274576 

22-8910 

567 

321489 

23-8118 

525 

275625 

22-9129 

568 

322624 

23-8327 

526 

276676 

22-9347 

569 

323761 

23-8537 

527 

277729 

22-9565 

570 

324900 

23-8747 

528 

278784 

22-9782 

571 

326041 

23-8956 

529 

279841 

23-0000 

572 

327184 

23-9165 

1  530 

280900 

23-0217 

573 

328329 

23-9374 

531 

281961 

23-0434 

574 

329476 

23-9583 

532 

283024 

2a-0651 

575 

330625 

23-9792 

533 

284089 

23-0868 

576 

331776 

24-0000 

534 

285156 

23-1084 

577 

332929 

24-0208 

535 

286225 

23-1301 

578 

334084 

24-0416 

536 

287296 

23-1517 

579 

335241 

24-0624 

537    288369 

23-1733 

580 

336400 

24-0832 

538    289444 

23-1948 

581 

337561 

24-1039 

539    290521 

23-2164 

582 

338724 

24-1247 

540 

291600 

23-2379 

583 

339889 

24-1454 

541 

292681 

23-2594 

584 

341056 

24-1661 

542 

293764 

23-2809 

585 

342225 

24-1868 

543 

294849 

23-3021 

586 

343396 

24-2074 

544 

295936 

23-3238 

587 

344569 

24-2281 

545 

297025 

23-3452 

588 

345744 

24-2487 

546 

298116 

23-3666 

589 

346921 

24-2693 

547 

299209 

23-3880 

590 

348100 

24-2899 

548 

300304 

23-4094 

591 

349281 

24-3105 

549 

301401 

2,°  4307 

592 

350464 

24-3310 

550 

302500 

i3-4521 

593 

351649 

24-3516 

551 

303601 

23-4734 

594 

352836 

24-3721 

552 

304704 

23-4947 

595 

354025 

24-3926 

553 

305809 

23-5159 

596 

355216 

24-4131 

554 

306916 

23-5372 

597 

356409 

24-4336 

555 

308025 

23-5584 

598 

357604 

24-4540 

556 

309136 

23-5796 

599 

358801 

24-4745 

557 

310249 

23-6008 

600 

360000 

24-4949 

558 

311364 

23-6220 

601 

361201 

24-5153 

559 

312481 

23-6432 

602 

362404 

24-5357 

101 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

603 

363609 

24-5560 

646 

417316 

25-4165 

604 

364816 

24-5764 

647 

418609 

25-4362 

605 

366025 

24-5967 

648 

419904 

25-4558 

606 

867236 

24-6171 

649 

421201 

25-4755 

607 

368449 

24-6374 

650 

422500 

25-4950 

1   608 

369664 

24-6576 

651 

423801 

25-5147 

609 

370881 

24-6779 

652 

425104 

25-5343 

610 

372100 

24-6982 

653 

426409 

25-5539 

611 

373321 

24-7184 

654 

427716 

25-5734 

612 

374544 

24-7386 

655 

429025 

25-5930 

613 

375769 

24-7588 

656 

430336 

25-6125 

614 

376996 

24-7790 

657 

431649 

25-6320 

615 

378225 

24-7992 

658 

432964 

25-6515 

616 

379456 

24-8193 

659 

434281 

25-6710 

617 

380689 

24-8395 

660 

435600 

25-6905 

618 

381924 

24-8596 

661 

436921 

25-7099 

619 

383161 

24-8797 

662 

438244 

25-7204 

620 

384400 

24-8998 

663 

439569 

25-7488 

621 

385641 

24-9199 

664 

440896 

25-7682 

622 

386884 

24-9399 

665 

442225 

25-7876 

623 

388129 

24-9600 

666 

443556 

25-8070 

624 

389376 

'  24-9800 

667 

444889 

25-8263 

625 

390625 

25-0000 

668 

446224 

25-8457 

626 

391876 

25-0200 

669 

447561 

25-8650 

627 

393129 

25-0400 

670 

448900 

25-8844 

628 

394384 

25-0600 

671 

450241 

25-9037 

629 

395641 

25-0799 

672 

451584 

25-9230 

630 

396900 

25-0998 

673 

452929 

25-9422 

631 

398161 

25-1197 

674 

454276 

25-9615 

632 

399424 

25-1396 

675 

455625 

25-9808 

633 

400689 

25-1595 

676 

456976 

26-0000 

634 

401956 

25-1794 

677 

458329 

26-0192 

635 

403225 

25-1992 

678 

459684 

26-0384 

636 

404496 

25-2190 

679 

461041 

26-0576 

637 

405769 

25-2389 

680 

462400 

26-0768 

638 

407044 

25-2587 

681 

463761 

26-0960 

639 

408321 

25-2785 

682 

465124 

26-1151 

640 

409600 

25-2982 

683 

466489 

26-1343 

641 

410881 

25-3180 

684 

467856 

26-1534 

642 

412164 

25-3377 

685 

469225 

26-1725 

643 

413449 

25-3574 

686 

470596 

26-1916 

644 

414736 

25-3772 

687 

471969 

26-2107 

645 

416025 

25-3968 

688 

473344 

26-2297 

!     ! 

i         ! 

102 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No.     Squares. 

Square  Roots. 

689 

474721 

26-2488 

7S2 

535824 

27-0555 

690 

476100 

26-2678 

733 

537289 

27-0740 

691 

477481 

26-2869 

734 

538756 

27-0924   i 

692 

478864 

26-3059 

735 

540225 

27-1109 

693 

480249 

26-3249 

736 

541696 

27-1293 

j  694 

481636 

26-3439 

737 

543169 

27-1477 

695 

483025 

26-3629 

738 

544644 

27-1662 

696 

484416 

26-3818 

739 

546121 

27-1846 

697 

485809 

26-4008 

740 

547600 

27-2029 

698 

487204 

26-4197 

741 

549081 

27-2213 

699 

488601 

26-4386 

742 

550564 

27-2397 

700 

490000 

26-4575 

743 

552049 

27-2580 

701 

491401 

26-4764 

744 

553536 

27-2764 

702 

492804 

26-4953 

745 

555025 

27-2947 

703 

494209 

26-5141 

746 

556516 

27-3130 

704 

495616 

26-533C 

747 

558009 

27-3313 

705 

497025 

26-5518 

748  I   559504 

27-3496 

706 

498436 

26-5707 

749 

561001 

27-3679 

707 

499849 

26-5895 

750 

562500 

27-3861 

708 

501264 

26-6083 

751 

564001 

27-4044 

709 

502681 

26-6271 

752 

565504 

27-4226 

710 

504100 

26-6458 

753 

567009 

27-4408 

711 

505521 

26-6646 

754 

568516 

27-4591 

712 

506944 

26-6833 

755 

570025 

27-4773 

713 

508369 

26-7021 

756 

571536 

27-4955 

714 

509796 

26-7208 

757  i   573049 

27-5136 

715 

511225 

26-7395    758  i   574564 

27-5318 

716 

512656 

26-7582    759  !   576081 

27-5500 

717 

514089 

26-7769    760    577600 

27-5681 

718 

515524 

26-7955    761    579121 

27-5862 

719 

516961 

26-8142  !  762    580644 

27-6043 

720 

518400 

26-8328 

763    582169 

27-6225 

721 

519841 

26-8514 

764 

583696 

27-6405 

722 

521284 

26-8701 

765 

585225 

27-6586 

723 

522729 

26-8887 

766 

586756 

27-6767 

724 

524176 

26-9072 

767    588289 

27-6948 

725 

525625 

26-9258 

768    589824 

27-7128 

726 

527076 

26-9444 

769    591361 

27-7308   i 

727 

528529 

26-9629 

770    592900 

27-7489   j 

728 

529984 

26-9815 

771    594441 

27-7669 

729 

531441 

27-0000 

772  !   595984 

27-7849 

730 

532900 

27-0185 

773 

597529 

27-8029 

731 

534361 

27-0370 

774 

599076 

27-820S 

103 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROO'i'S 

OF  NUMBERS.—  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

775 

600625 

27-8388 

818 

669124 

28-6007 

776 

602176 

27-8568 

819 

670761 

28-6182 

777 

603729 

27-8747 

820 

672400 

28-6356 

778 

605284 

27-8926 

821 

674041 

28-6531 

!  779 

606841 

27-9106 

822 

675684 

28-6705 

780 

608400 

27-9285 

823 

677329 

28-6880 

781 

609961 

27-9464 

824 

678976 

28-7054 

782 

611524 

27-9643 

825 

680625 

28-7228 

783 

613089 

27-9821 

826 

682276 

28-7402 

784 

614656 

28-0000 

827 

683929 

28-7576 

785 

616225 

28-0179 

828 

685584 

28-7750 

786 

617796 

28-0357 

829 

687241 

28-7924 

787 

619369 

28-0535 

830 

688900 

28-8097 

788 

620944 

28-0713 

831 

690561 

28-8271 

789 

622521 

28-0891 

832 

692224 

28-8444 

i  790 

624100 

28-1069 

833 

693889 

28-8617 

791 

625681 

28-1247 

834 

695556 

28-8791 

792 

627264 

28-1425 

835 

697225 

28-8964 

793 

628849 

28-1603 

836 

698896 

28-9137 

794 

630436 

28-1780 

837 

700569 

28-9310   i 

795 

632025 

28-1957 

838 

702244 

28-9482 

!  796 

633616 

28-2135 

839 

703921 

28-9655 

797 

635209 

28-2312 

840 

705600 

28-9828 

798 

636804 

28-2489 

841 

707281 

29-0000 

799 

638401 

28-2666 

842 

708964 

29-0172 

800 

640000 

28-2843 

843 

710649 

29-0345 

801 

641601 

28-3019 

844 

712336 

29-0517 

802 

643204 

28-3196 

845 

714025 

29-0689 

803 

644809 

28-3373 

846 

715716 

29-0861 

804 

646416 

28-3549 

847 

717409 

29-1033 

1   805 

648025 

28-3726 

848 

719104 

29-1204 

806 

649636 

28-3901 

849 

720801 

29-1376 

807 

651249 

28-4077 

850 

722500 

29-1548 

808 

652864 

28-4253 

851 

724201 

29-1719 

809 

654481 

28-4429 

852 

725904 

29-1890 

810 

656100 

28-4605 

853 

727609 

29-2062 

811 

657721 

28-4781 

854 

729316 

29-2233 

812 

659344 

28-4956 

855 

731025 

29-2404 

813 

660969 

28-5132 

856 

732736 

29-2575 

814 

662596 

28-5307 

857 

734449 

29-2746 

815 

664225 

28-5482 

858 

736164 

29-2916 

816 

665856 

28-5657 

859 

737881 

29-3087 

817 

667489 

28-5832 

860 

739600 

29-3258 

r=--                          _  —  -  • 

104 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS  : 

OF  NUMBERS.—  CONTINUED.   • 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

!   861 

741321 

29-3428 

904 

817216 

30-0666 

862 

743044 

29-3598 

905 

819025 

30-0832 

863 

744769 

29-3769 

906 

820836 

30-0998 

864 

746496 

29-3939 

907 

822649 

30-1164 

865 

748225 

29-4109 

908 

824464 

30-1330 

866 

749956 

29-4279 

909 

826281 

30-1496 

867 

751689 

29-4449 

910 

828100 

30-1662 

868 

753424 

29-4618 

911 

829921 

30-1828 

869 

755161 

29-4788 

912 

831744 

30-  1993 

870 

756900 

29-4958 

913 

833569 

30  2159 

871 

758641 

29-5127 

914 

835396 

30-2324 

872 

760384 

29-5296 

915 

837225 

30-2490 

873 

762129 

29-5466 

916 

839056 

30-2655 

874 

763876 

29-5635 

917 

840889 

30-2820 

875 

765625 

29-5804 

918 

842724 

30-2985 

876 

767376 

29-5973 

919 

844561 

30-3150 

877 

769129 

29-6142 

920 

846400 

30-3315 

878 

770884 

29-6311 

921 

848241 

30-3480 

879 

772641 

29-6479 

922 

850084 

30-3645 

880 

774400 

29-6648 

923 

851929 

30-3809 

881 

776161 

29-6816 

924 

853776 

30-3974 

882 

777924 

29-6985 

925 

855625 

30-4138 

883 

779689 

29-7153 

926 

857476 

30-4302 

884 

781456 

29-7321 

927 

859329 

30-4467 

885 

783225 

29-7489 

928 

861184 

30-4631 

886 

784996 

29-7658 

929 

863041 

30-4795 

887 

786769 

29-7825 

930 

864900 

30-4959 

888 

788544  • 

29-7993 

931 

866761 

30-5123 

889 

790321 

29-8161 

932 

868624 

30-5287 

890 

792100 

29-8329 

933 

870489 

30-5450 

891 

793881 

29-8496 

934 

872356 

30-5614 

892 

795664 

29-8664 

935 

874225 

30-5778   j 

893 

797449 

29-8831 

936 

876096 

30-5941 

894 

799236 

29-8998 

937 

877969 

30-6105 

895 

801025 

29-9166 

938 

879844 

30-6268 

896 

802816 

29-9333 

939 

881721 

30-6431 

897 

804609 

29-9500 

940 

883600 

30-6594 

1  898 

806404 

29-9666 

941 

885481 

30"i757 

999 

808201 

29-9833 

942 

887364 

30-6920 

900 

810000 

30-0000 

943 

889249 

30-7083 

901 

811801 

30-0167 

944 

891136 

30-7246 

'  902 

813604 

30-0333 

945 

893025 

30-7409 

903 

815409 

30-0500 

946 

894916 

30-7571 

105 

A  TABLE  OF  THE  SQUARES  AND  SQUARE  ROOTS 

OF  NUMBERS.  —  CONTINUED. 

From  1  to  1000. 

No. 

Squares. 

Square  Roots. 

No. 

Squares. 

Square  Roots. 

947 

896809 

30-7734 

974 

948676 

31-2090 

948 

898704 

30-7896 

975 

950625 

31-2250 

949 

900601 

30-8058  ' 

976 

952576 

31-2410 

950 

902500 

30-8221 

977 

954529 

31-2570 

951 

904401 

30-8383 

978 

956484 

31-2730 

952 

906304 

30-8545 

979 

958441 

31-2890 

953 

908209 

30-8707 

980 

960400 

31-3050 

954 

910116 

30-8869 

981 

962361 

31-3209 

955 

912025 

30-9031 

982 

964324 

31-3369 

956 

913936 

30-9192 

983 

966289 

31-3528 

957 

915849 

30-9354 

984 

968256 

31-3688 

958 

917764 

30-9516 

985 

970225 

31-3847 

959 

919681 

30-9677 

986 

972196 

31-4006 

960 

921600 

30-9839 

987 

974169 

31-4166 

961 

923521 

31-0000 

988 

976144 

31-4325 

962 

925444 

31-0161 

989 

978121 

31-4484 

963 

927369 

31-0322 

990 

980100 

31-4643 

964 

929296 

31-0483 

991 

982081 

31-4802 

965 

931225 

31-0644 

992 

984064 

31-4960 

966 

933156 

31-0805 

993 

986049 

31-5119   ! 

967 

935089 

31-0966 

994 

988036 

31-5278   , 

968 

937024 

31-1127 

995 

990025 

31-5436 

969 

938961 

31-1288 

996 

992016 

31-5595 

970 

940900 

31-1448 

997 

994009 

31-5753   i 

971 

942841 

31-1609 

998 

996004 

31-5911 

972 

944784 

31-1769 

999 

998001 

31-6070 

973 

946729 

31-1929 

1000 

1000000 

31-6228 

106 

TABLE  OF  SLOPES,  &c.—  For  TOPOGRAPHY. 

Vertical  Rise 

Horizontal 

Vertical  Rise 

Horizontal 

Degrees. 

in  100  feet 
horizontal. 

Distance  to  a 
rise  of  10  feet. 

Degrees. 

in  100  feet 
horizontal. 

Distance  to  a 
rise  of  10  feet. 

0 

1 

1-75 

572-9 

o 

19 

34-43 

29-0 

2 

3-49 

286:4 

20 

36-40 

27-5 

3 

5-24 

190-8 

21 

38-40 

26-0 

4 

6-99 

143-0 

22 

40-40 

24-7 

5 

8-75 

114-3 

23 

42-45 

23-5 

6 

10-51 

95-1 

24 

44-52 

22-4 

7 

12-28 

81-4 

25 

46-63 

21-4 

8 

14-05 

71-2 

26 

48-77 

20-5 

9 

15-83 

63-1 

27 

50-95 

19-6 

10 

17-63 

56-7 

28 

53-17 

18-8 

11 

19-44 

51-4 

29 

55-43 

18-0 

12 

21-25 

47-0 

30 

57-73 

17-3 

13 

23-09 

43-3 

35 

70-02 

14-2 

14 

24-93 

40-1 

40 

83-91 

11-9 

15 

26-79 

37-3 

45 

100-00 

10-0 

16 

28-67 

34-9 

50 

119-17 

8-4 

17 

30-57 

32-7 

55 

142-81 

7-0 

]8 

32-49 

30-7 

60 

173-20 

5-7 

;    €1^     ! 

R.  B.  WEARS,  STEREOTYPER. 

AN  INITIAL  FINE  OF  25  CENTS 

OVERDUE. 


p 


YB  5360! 


2/y 


W~  •-•• 

_ 


